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Studies of dense plasmas in laser generated shock wave experiments Parfeniuk, Dean Allister 1982

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AN INVESTIGATION OF THE EFFECT OF CONDUCTING SHELLS ON ELECTRODELESS BREAKDOWN by DEAN ALLISTER PARFENIUK B. A. Sc., The University of B r i t i s h Columbia, 1981 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF APPLIED SCIENCE in THE FACULTY OF GRADUATE STUDIES (DEPARTMENT OF PHYSICS) We accept t h i s thesis as conforming to the required standard THE © UNIVERSITY OF BRITISH COLUMBIA November 1982 Dean A l l i s t e r Parfeniuk, 1982 I n p r e s e n t i n g t h i s t h e s i s i n p a r t i a l f u l f i l m e n t o f t h e r e q u i r e m e n t s f o r an advanced degree a t the U n i v e r s i t y o f B r i t i s h C o l u m b i a , I agree t h a t t h e L i b r a r y s h a l l make i t f r e e l y a v a i l a b l e f o r r e f e r e n c e and s t u d y . I f u r t h e r a gree t h a t p e r m i s s i o n f o r e x t e n s i v e c o p y i n g o f t h i s t h e s i s f o r s c h o l a r l y purposes may be g r a n t e d by t h e head o f my department o r by h i s o r h e r r e p r e s e n t a t i v e s . I t i s u n d e r s t o o d t h a t c o p y i n g o r p u b l i c a t i o n o f t h i s t h e s i s f o r f i n a n c i a l g a i n s h a l l n o t be a l l o w e d w i t h o u t my w r i t t e n p e r m i s s i o n . Department o f PHYSICS The U n i v e r s i t y o f B r i t i s h C o l u m b i a 1956 Main M a l l V a n c o u v e r , Canada V6T 1Y3 Date 0 . , . ICs/fi-Z. »E-6 C3/81) i i ABSTRACT Two theoretical models were developed to predict the eff e c t of conducting she l l s on electrodeless breakdown at low frequencies ( < 1 kHz). If glass s h e l l s containing gases at low pressures are immersed in an alternating e l e c t r i c f i e l d , of s u f f i c i e n t magnitude, these bulbs w i l l emit pulses of l i g h t . Together the th e o r e t i c a l models predict the form of the c a l i b r a t i o n curves of these bulbs, which are plots of the pulse rate as a function of the magnitude of the applied f i e l d . Experimental v e r i f i c a t i o n of these models was observed. Furthermore, the surface conductivity of paxolin and neoprene was measured as a function of frequency in the range 40 Hz to 1 kHz. TABLE OF CONTENTS Page ABSTRACT i i TABLE OF CONTENTS i i i LIST OF TABLES v i LIST OF FIGURES • v i i ACKNOWLEDGEMENTS ix CHAPTER 1 INTRODUCTION 1 CHAPTER 2 THEORY 5 2.1 BASIC MECHANISM 5 2.2 GENERAL THEORY WITH CONDUCTIVITY INCLUDED 14 2.3 THEORY PARTICULARIZED TO PROLATE SPHEROIDAL COORDINATES 22 2.4 THE CYLINDRICAL LIMIT OF A SPHEROID 33 2.5'APPROXIMATE SOLUTION OF THE "SWITCH ON" PROBLEM WITH THE APPLIED FIELD PERPENDICULAR TO THE AXIS OF THE SPHEROID 37 2.6 APPROXIMATE SOLUTION OF THE "SWITCH ON" PROBLEM WITH THE APPLIED FIELD PARALLEL TO THE AXIS OF THE SPHEROID 47 2.7 APPROXIMATE SOLUTION OF THE "SWITCH ON" PROBLEM IN THE SPHERICAL LIMIT WITH THE APPLIED FIELD IN THE X-DIRECTION 49 2.8 APPROXIMATE SOLUTION OF THE "SWITCH ON" PROBLEM IN THE SPHERICAL LIMIT WITH THE APPLIED FIELD IN THE Z-DIRECTION 52 2.9 EXACT SOLUTION OF THE "SWITCH ON" PROBLEM FOR A SPHERE WITH THE APPLIED FIELD IN THE Z-DIRECTION 54 2.10 EXACT SOLUTION OF THE "SWITCH ON" PROBLEM FOR A CYLINDER WITH THE APPLIED FIELD PERPENDICULAR TO ITS AXIS 57 2.11 RESPONSE OF A CONDUCTING SHELL TO AN ALTERNATING ELECTRIC FIELD 60 2.12 "RESET SCREENING" MODEL 64 i v 2.13 LINKING OF THE "RESET SCREENING" AND "STATIC SCREENING" MODELS 72 CHAPTER 3 EXPERIMENTS 75 3.1 INTRODUCTION 75 3.2 VERIFICATION OF THE "STATIC SCREENING" MODEL 78 3.3 EXPERIMENTAL VERIFICATION OF THE FREQUENCY DEPENDENCE OF THE CONDUCTIVITY OF NEOPRENE 94 3.4 EXPERIMENTAL VERIFICATION OF THE THEORY WHICH EXPLAINS THE FORM OF THE CALIBRATION CURVES OF THE BULBS 105 CHAPTER 4 CONCLUSIONS 116 4.1 INTRODUCTION 116 4.2 CONCLUSIONS 117 BIBLIOGRAPHY ..122 ' APPENDIX A 123 A.1 MOMENT OF EQUATION 2.3-19 FOR THE Z-DIRECTION PROBLEM 123 A. 2 MOMENT OF EQUATION 2.3-19 FOR THE X-DIRECTION PROBLEM 125 APPENDIX B 1 28 B. 1 SOLUTION OF THE X-DIRECTION PROBLEM FOR AN ASYMPTOTICALLY LONG SPHEROID 128 B.2 1-DIMENSIONAL EIGENVALUE PROBLEM 132 B. 3 3-DIMENSIONAL EIGENVALUE PROBLEM 133 APPENDIX C 136 C. 1 SOLUTION OF THE Z-DIRECTION PROBLEM FOR AN ASYMPTOTICALLY LONG SPHEROID 136 C.2 1-DIMENSIONAL EIGENVALUE PROBLEM 141 C.3 3-DIMENSIONAL EIGENVALUE PROBLEM 142 V APPENDIX D 1 49 D. 1 SOLUTION OF THE X-DIRECTION "SWITCH ON" PROBLEM FOR A SPHEROID TAKEN TO THE SPHERICAL LIMIT 149 APPENDIX E 1 54 E. 1 SOLUTION OF THE Z-DIRECTION "SWITCH ON" PROBLEM FOR A SPHEROID TAKEN TO THE SPHERICAL LIMIT 154 v i LIST OF TABLES Page TABLE I, STATIC SCREENING DATA 88 TABLE II, CONDUCTIVITY OF PAXOLIN AND NEOPRENE 94 TABLE III CONDUCTIVITY OF NEOPRENE (RUN WITH 3 SHEETS) ..100 TABLE IV, CONDUCTIVITY OF NEOPRENE (RUN WITH 5 SHEETS) ..100 TABLE V, BULB DATA 107 TABLE VI, EXPERIMENTALLY OBTAINED TIME CONSTANTS 112 TABLE VII EXPERIMENTALLY OBTAINED PHASE ANGLES 114 v i i LIST OF FIGURES Page FIGURE 1, TYPICAL BULB 8 FIGURE 2, FIELD INTERNAL TO THE BULB 8 FIGURE 3, CALIBRATION CURVE FOR A NON-CONDUCTING BULB ....10 FIGURE 4, PHASE OF THE PULSES AT THRESHOLD FOR A NON-CONDUCTING BULB 10 FIGURE 5, SAMPLE CALIBRATION CURVE 13 FIGURE 6, RESULTS OF THE PROPOSED THEORETICAL MODELS 13 FIGURE 7, ELLIPTICAL COORDINATES 23 FIGURE 8, PROLATE SPHEROIDAL COORDINATES 23 FIGURE 9, THEORETICAL 6 (r*,t)/e0E Cos* PLOTTED AGAINST Qt/Re0 1 X . .x. 41 FIGURE 10,THEORETICAL 5 ( r , t )/e 0 E Cos • PLOTTED AGAINST flt/Re0 3.X 42 FIGURE 11,THEORETICAL 6 ( r , t ) / c 0 E Cos t PLOTTED AGAINST Gt/Re0 5.X. 43 FIGURE 12,THEORETICAL « ( r , t ) / t 0 E Cos • PLOTTED AGAINST Qt/Re0 7.X ; 44 FIGURE 13,DEMONSTRATION OF THE CONVERGENCE OF 6 ,6 AND 6 3 X . . ? X 45 7 X FIGURE 14,DEMONSTRATION OF THE CONVERGENCE OF 6 TO 6 7.x 46 FIGURE 15,SPHERICAL COORDINATES 55 FIGURE 16,CYLINDRICAL COORDINATES 55 FIGURE 17,DEFINITION OF THE RADIUS OF A SPHEROID 59 FIGURE 18,DEFINITION OF PHASE SHIFT 63 FIGURE 19,DEFINITION OF "a" FOR A TRIANGULAR WAVE 70 FIGURE 20,DEFINITION OF "a" FOR A CLIPPED TRIANGULAR WAVE 70 FIGURE 21,RESULTS OF THE "STATIC" AND "RESET SCREENING" MODELS . 73 FIGURE 22,DEVICE FOR PRODUCING ELECTRIC FIELDS 76 v i i i FIGURE 23,DEVICE FOR MEASURING ELECTRIC FIELDS 77 FIGURE 24,TUBE ORIENTATION WITH RESPECT TO CARTESIAN COORDINATES 79 FIGURE 25,A PLOT OF THE THEORETICALLY PREDICTED XM/Xx 82 FIGURE 26,CONFIGURATION OF THE APPARATUS USED TO MEASURE (EA/E, )x 85 FIGURE 27,CONFIGURATION OF THE APPARATUS USED TO MEASURE (EA/E, )„ 86 FIGURE 28,A PLOT OF THE THEORETICAL AND EXPERIMENTAL X„ A i 89 FIGURE 29,CONDUCTIVITY OF PAXOLIN PLOTTED AGAINST FREQUENCY 92 FIGURE 30,CONDUCTIVITY OF NEOPRENE PLOTTED AGAINST FREQUENCY 93 FIGURE 31,NEOPRENE DIMENSIONS '. 95 FIGURE 32,CONFIGURATION OF THE APPARATUS USED TO MEASURE fl/e 98 FIGURE 3.3,EXPERIMENTAL e/c 0, FOR NEOPRENE, PLOTTED AGAINST FREQUENCY 102 FIGURE 34 /EXPERIMENTAL n, FOR NEOPRENE, PLOTTED AGAINST FREQUENCY 103 FIGURE 35,EXPERIMENTAL Q/c, FOR NEOPRENE, PLOTTED AGAINST FREQUENCY 104 FIGURE 36,SODA-LIME BULB CALIBRATION CURVE FOR 40 Hz 108 FIGURE 37,SODA-LIME BULB CALIBRATION CURVE FOR 60 Hz 109 FIGURE 38,SODA-LIME BULB CALIBRATION CURVE FOR 80 Hz 110 FIGURE 39,SODA-LIME BULB CALIBRATION CURVE FOR 100 Hz ...111 FIGURE 40,PHASE SHIFT OF THE BULB'S PULSES AT THRESHOLD 115 FIGURE 41 ,RESULTS OF THE PROPOSED THEORETICAL MODELS 118 FIGURE 42,PHASE OF THE PULSES AT THRESHOLD FOR A NON-CONDUCTING BULB 118 FIGURE 43,APPROXIMATE FORM OF 6, 145 FIGURE 44,EMPIRICAL FIT OF * z 147 ix ACKNOWLEDGEMENTS I would l i k e to thank my supervisor Dr. F. L. Curzon for suggesting t h i s project and for his supervision. His suggestions were most useful in both the experimental and th e o r e t i c a l aspects of t h i s project. I would also l i k e to thank A. Cheuck for his help with the experimental apparatus, L. Da Si l v a for his help with the computer programming and M. Feeley for many useful discussions. - 1 -CHAPTER 1 INTRODUCTION In the past few years, there has been concern about the possible e f f e c t s of strong, low frequency, e l e c t r i c f i e l d s on an individual's health. Those who may be concerned include employees of u t i l i t y companies and individuals who l i v e near transmission l i n e s since these people receive long term exposures to these f i e l d s . Hence, i t i s desired to develop devices to monitor e l e c t r i c f i e l d doses. These devices could be used in studies directed towards determining the health hazards of these f i e l d s . A l l currently e x i s t i n g e l e c t r i c f i e l d monitors are based on current induction. For example, the primary part of most devices i s a p a r a l l e l plate capacitor. The electronics required to measure the induction and convert the result into an estimate of the e l e c t r i c f i e l d i s located between the capacitor plates. Clearly these devices have many disadvantages. For example, the s e n s i t i v i t y of the device depends on i t s orientation with respect to the f i e l d . The accuracy i s affected by harmonic d i s t o r t i o n s in the f i e l d . These devices consist of metal parts. This can result in users of the device receiving e l e c t r i c a l shocks. It has been found that the magnitude of alternating e l e c t r i c f i e l d s can be measured with a device based on e l e c t r i c a l breakdown of .gas inside . insulating s h e l l s . - 2 -Harries 1 established that e l e c t r i c a l breakdown occurs when glass bulbs f i l l e d with gas at low pressures are exposed to alternating e l e c t r i c f i e l d s of s u f f i c i e n t magnitude. This i s known as an "electrodeless breakdown". It turns out the rate of breakdown occurrence i s proportional to the magnitude of the e l e c t r i c f i e l d and the frequency. These bulbs can be ca l i b r a t e d so that the magnitude of an e l e c t r i c f i e l d can be determined from the breakdown rate of a bulb. An e l e c t r i c f i e l d monitor based on t h i s phenomena i s currently being developed at the University of B r i t i s h 5 Columbia. This sort of device has many advantages over a device based on capacitive coupling. For example, i f the glass bulb i s spherical, the breakdown rate is. independent of the orientation of the bulb with respect to the f i e l d . Furthermore the breakdown rate of the bulb i s not subs t a n t i a l l y affected by small harmonic d i s t o r t i o n s in the e l e c t r i c f i e l d . Each breakdown results in a pulse of l i g h t , which can be transmitted from the v i c i n i t y of the bulb to the pulse counting device with a nonconducting o p t i c a l f i b e r . Thus the ele c t r o n i c s (pulse counter) can be shielded without a f f e c t i n g the f i e l d near the bulb. This reduces the ele c t r o n i c noise problem. Furthermore, t h i s device i s d i r e c t l y compatible with d i g i t a l processing. F i n a l l y , the detection device contains no metal parts, hence the chances of a user receiving an e l e c t r i c a l shock are minimized. In t h i s thesis, the results of research directed 1 Harries,w.L. and Von Engel,A., Proc. Phys. S o c , London, Sect. B64,915(1951) - 3 -towards understanding the operation of the bulbs are presented. In p a r t i c u l a r , the e f f e c t of the conductivity of the bulb's s h e l l was studied. This ef f e c t i s important since the conductivity of the s h e l l can change due to the cleanliness of the glass and humidity. This a f f e c t s the accuracy of the monitor. Theoretical models were developed to predict the effect of the conduction and experiments were devised to test these models. Chapter 2 contains the derivations of the th e o r e t i c a l models. It turns out two models were required to explain the operation of a bulb with a conducting s h e l l . These models are c a l l e d ; the " s t a t i c screening" model and the "reset screening" model. These models predict the frequency of breakdown, f B , as a function of the applied f i e l d strength, E A , and frequency, f A . The results predict the form of the c a l i b r a t i o n curves, which are plots of f B versus E A at fixed f A . The mathematical d e t a i l s appear in the appendices at the end of the t h e s i s . The experimental results are presented in chapter 3. In section 1 the experimental apparatus i s described. Next the " s t a t i c screening" model was tested d i r e c t l y . The results are presented in section 2. This was accomplished by measuring the e l e c t r i c f i e l d inside conducting tubes. A consequence of t h i s experiment was that the conductivity of neoprene was measured as a function of frequency in the range 0 Hz to 1kHz. The frequency dependence of the conductivity of neoprene was measured with a d i f f e r e n t technique and the - 4 -results are given in section 3. This was accomplished by measuring the magnitudes and phases of the voltage across neoprene sheets and the current through the sheets. These results agreed with those from the previous section. F i n a l l y , the " s t a t i c screening" model and "reset screening" model were tested simultaneously. The results are presented in section 4. The th e o r e t i c a l " s t a t i c screening curve" and the the o r e t i c a l "reset screening curve" were f i t to a t y p i c a l c a l i b r a t i o n curve. The parameters obtained from t h i s f i t were used to calculate the conductivity of the glass in two independent ways. It turned out these values were consistent. Hence these models were determined to be v a l i d . Chapter 4 i s a summary of the main results of the thesis, emphasizing their p r a c t i c a l s i g n i f i c a n c e . - 5 -CHAPTER 2 THEORY 2.1 BASIC MECHANISM It has been established by Harries 1 that glass bulbs containing gases at pressures in a s p e c i f i c regime would emit pulses of li g h t when placed in an alternating e l e c t r i c f i e l d of s u f f i c i e n t magnitude. Furthermore, i t was found the pulse rate was approximately proportional to the magnitude and frequency of the e l e c t r i c f i e l d . An explanation of these findings i s given below with the assumption that the bulb's glass s h e l l i s nonconducting. This means the e l e c t r i c f i e l d inside the bulb is e s s e n t i a l l y unaffected by the s h e l l . A t y p i c a l bulb i s shown in figure 1. To explain the operation of a bulb, suppose that i n i t i a l l y there is no external or internal e l e c t r i c f i e l d s . As the external f i e l d i s increased, the internal f i e l d also increases. A free electron in t.he bulb tends to move in the opposite d i r e c t i o n to the f i e l d inside the bulb. As i t travel s along, i t i s affected by c o l l i s i o n s with gas atoms in the bulb. The average distance an electron travels between c o l l i s i o n s i s the thermal mean free path which w i l l be denoted XT. The mean free path i s much smaller than the dimensions of the bulb. If the e l e c t r i c f i e l d i s below the threshold e l e c t r i c f i e l d E 0 , where the threshold f i e l d i s defined as the minimum f i e l d which causes the bulb to emit pulses of l i g h t , the free electrons in the bulb do not gain - 6 -enough energy to ionize the gas molecules on impact over the dimension of the bulb. There are two mechanisms at f i e l d s below threshold which prevent electrons from becoming energetic enough to ionize gas molecules. F i r s t l y , the gas molecules absorb a fractio n of the free electron's energy due to thermal c o l l i s i o n s which occur at an average distance XT apart. Secondly, the dimension of the bulb i s too small to allow the electron to gain enough energy, over this distance, to reach the energy required to ionize a gas molecule on c o l l i s i o n . One can say the mean free path for ionization i s greater than the dimension of the bulb, where the mean free path for ionization, which w i l l be denoted A,, is defined as the average distance an electron must travel in order to gain enough energy to ionize a gas molecule. As the magnitude of the e l e c t r i c f i e l d i s increased, X, w i l l decrease since the thermal c o l l i s i o n s only take a fraction of a free electron's energy. With large enough e l e c t r i c f i e l d s , X, w i l l become smaller than the dimension of the bulb. Under these conditions, the electron population w i l l approximately double every time the electrons move a distance equal to X,. Hence the electron population w i l l grow exponentially. This phenomenon i s known as an electron avalanche. The number of electrons involved in an electron avalanche i s : N - 2 L / X ' 2 . 1 - 1 Here N i s the number of electrons involved in the avalanche and L i s the length of the avalanche which is approximately 7 -equal to the dimension of the bulb. However, the occurrence of these avalanches does not mean threshold has been reached, since in order to sustain the ionization process, each avalanche must produce at least one secondary electron. This secondary electron w i l l produce another avalanche, hence the ionization process is sustained. These secondary electrons can be produced by several mechanisms. F i r s t l y , secondary electrons may be produced by posit i v e ions s t r i k i n g the wall of the bulb. Secondly, electrons involved in the avalanche may produce photons upon s t r i k i n g the bulb's wall which in turn could produce secondary electrons by ionizing gas molecules. If Y i s the pr o b a b i l i t y that one electron ion pair w i l l produce a secondary electron, then the requirement for a s e l f - s u s t a i n i n g ionization process i s : 1/Y.2WX, 2.1-2 The minimum e l e c t r i c f i e l d required to obtain a se l f - s u s t a i n i n g ionization process i s the threshold f i e l d E 0. A, and Y are a function of gas pressure and the e l e c t r i c f i e l d . Hence equation 2.1-2 may be inverted in p r i n c i p l e to obtain an equation of the form: E 0= F(pL)/L 2 > 1 _ 3 This i s known as Paschen's Law. Values of F may be obtained from tables, hence E 0 may be calculated. - 8 -TYPICAL BULB ^ GLASS S H E L L GAS CONTENT IS ARGON AT ONE TORR I J v / F U L L S C A L E FIGURE 1 FIELD INTERNAL TO THE BULB FIGURE 2 - 9 -This entire breakdown process i s known as the Townsend Mechanism. 2 Many of the ions involved in the electron avalanche w i l l have been excited to higher energy states. These ions w i l l de-excite to more stable energy states thus emitting photons, hence a pulse of l i g h t i s formed. The ionization process w i l l continue u n t i l the f i e l d in the bulb i s approximately cancelled by the f i e l d created from the electrons and ions c o l l e c t i n g on the walls of the bulb. The time required for the avalanche to occur and the f i e l d to be reset i s assumed to be small compared to the period of the applied f i e l d . As the e l e c t r i c f i e l d external to the bulb i s further increased, the f i e l d internal to the bulb w i l l also increase u n t i l threshold i s again reached, then another avalanche w i l l occur emitting another pulse of li g h t and again resetting the bulb's internal f i e l d to zero. This process continues i n d e f i n i t e l y . See figure 2 for a graph of the internal f i e l d versus time. It i s evident that the frequency of the breakdown, f B , i s proportional -to the integer number of times the applied f i e l d E A , which i s defined in figure 2, can be divided by the threshold f i e l d , E 0 . Moreover, i t i s clear that f B i s proportional to the frequency of the applied f i e l d . Furthermore, there w i l l be two breakdowns per cycle at threshold, one at the extreme maximum of the f i e l d and one at the extreme minimum. Hence: 2 Meek,J.M. and Craggs,J.D., E l e c t r i c a l Breakdown of Gases , John Wiley and Sons, New York(1978) - 10 -CALIBRATION CURVE FOR A NON-CONDUCTING BULB 4 o A P P L I E D E L E C T R I C F I E L D FIGURE 3 PHASE OF THE PULSES AT THRESHOLD FOR A NON-CONDUCTING BULB FIGURE 4 - 1 1 -2.1-4 where u i s the frequency of the applied f i e l d and int( ) indicates the quantity in the brackets i s rounded down to the nearest integer. This breakdown process would imply the c a l i b r a t i o n curve should be a stepped curve as shown in figure 3. The c a l i b r a t i o n curve i s a plot of f B versus the applied f i e l d , E A . In fact, not a l l curves are stepped, some are quite l i n e a r . There are two mechanisms which cause t h i s . F i r s t l y , the s t a t i s t i c s of avalanche breakdowns smooth the curve somewhat.2 Secondly, the path of the breakdowns may be changing. For example, the f i r s t breakdown may occur in the centre of the bulb. Thus the f i e l d w i l l be reset in the centre of the bulb, but only p a r t i a l l y reset away from the centre. Hence the next breakdown w i l l occur away from the centre of the bulb at an apparently reduced threshold. This apparent changing of the threshold f i e l d i s also s t a t i s t i c a l in nature hence i t tends to smooth the c a l i b r a t i o n curve. See figure 3 for an experimentally obtained c a l i b r a t i o n curve. This curve w i l l be c a l l e d the "zero conductivity curve." Furthermore at threshold, where f B = W/TT, t h i s theory suggests the pulses should occur at the extreme maximum and minimum e l e c t r i c f i e l d as shown in figure 4. This configuration w i l l be defined to be a phase s h i f t of zero degrees. For some bulbs, the above theory explains their operation adequately, but for many bulbs the c a l i b r a t i o n - 12 -curves are displaced from the o r i g i n as shown in figure 5 . Furthermore, at threshold, the pulses have a non-zero phase s h i f t . The above theory demands the c a l i b r a t i o n curve goes through the o r i g i n and pulses have zero phase s h i f t at threshold. Therefore t h i s theory in i t s present form does not explain the observations shown in figure 5 . This problem can be overcome i f the bulb's s h e l l i s allowed to be s l i g h t l y conducting, where s l i g h t l y conducting means the conductivity of the s h e l l w i l l noticeably a f f e c t the f i e l d internal to the bulb, E, , with respect to the external f i e l d , E A . Current flow in the glass modifies the charge d i s t r i b u t i o n on the bulb surface. Two models are required to explain the observed r e s u l t s , each being v a l i d in a d i f f e r e n t regime of the bulb's operation. Both models contain one parameter, the conductivity of the surface, so from one c a l i b r a t i o n curve these models can be applied to obtain the conductivity of the s h e l l in two independent ways, thereby testing the self-consistency of the models. Furthermore, this conductivity can account for the phase s h i f t of the pulses at threshold. The f i r s t model, the " s t a t i c screening" model, i s v a l i d below threshold and possibly s l i g h t l y above threshold, but i s c e r t a i n l y not v a l i d when f B i s much greater than i t s threshold value. It turns out that t h i s model predicts E, w i l l be reduced and phase s h i f t e d with respect to E A. Hence, t h i s model predicts a reduction in the slope of the c a l i b r a t i o n curve as well as a phase s h i f t of the pulses at - 13 -SAMPLE CALIBRATION CURVE FOR A BULB 4 CD % L U 1-< or ZERO CONDUCTIV ITY CURVE PULSE OBSERVED ^ CURVE CAL IBRAT ION A P P L I E D E L E C T R I C F I E L D , E A — • FIGURE 5 RESULTS OF THE PROPOSED THEORETICAL MODELS OBSERVED CAL IBRAT ION A P P L I E D F I E L D , E A FIGURE 6 - 14 -threshold. These ef f e c t s are a function of the conductivity of the s h e l l . Note, since t h i s model i s v a l i d up to and possibly s l i g h t l y past threshold, at least the f i r s t point on the c a l i b r a t i o n curve must l i e on the curve predicted by t h i s model. This curve i s shown in figure 6. It w i l l be c a l l e d the " s t a t i c screening curve". The second model, the "reset screening" model, predicts the form of the c a l i b r a t i o n curve at high count rates. The th e o r e t i c a l curve predicted by th i s model i s shown in figure 6 and w i l l be c a l l e d the "reset screening curve". At high count rates, the th e o r e t i c a l curve i s p a r a l l e l to the "zero conductivity curve" but offset from i t as expected. Moreover, the offset of the "reset screening curve" from the "zero conductivity curve" depends on the conductivity of the glass. There exist two ways of c a l c u l a t i n g the conductivity of the glass with these models. F i r s t l y , the conductivity can be calculated from the r a t i o of the slopes of the "reset screening curve" to the " s t a t i c screening curve". Secondly, the conductivity can be calculated from the offset of the "reset screening curve" from the "zero conductivity curve". There i s one parameter in these models and two ways of measuring i t . Thus self-consistency can be demonstrated. 2.2 GENERAL THEORY WITH CONDUCTIVITY INCLUDED F i r s t l y the theory for the " s t a t i c screening" model is derived. The assumptions required for th i s . model are; the - 15 -frequency . of the applied f i e l d i s low so magnetic e f f e c t s can be neglected, current can only flow on the s h e l l (no avalanche breakdowns in the gas) and the s h e l l ' s conductivity is isotropic and Ohmic. Consider a closed s h e l l of a r b i t r a r y shape. Let the s h e l l be immersed in an external e l e c t r i c f i e l d . Furthermore, l e t the s h e l l possess a surface charge density 6. Assume the frequency of the applied f i e l d i s low so magnetic e f f e c t s can be neglected. Then the e l e c t r i c potential at a point on the s h e l l due to the surface charge density and the external f i e l d i s : * < r 0 > - </> T § T - 5 - + * E ( * ; > 2 > 2 „ 1 Here <£E()t0) i s the e l e c t r i c potential due to the external f i e l d at XD, <J>($0) is the t o t a l e l e c t r i c potential at X*0 and R i s the distance between 5t0 and the i n f i n i t e s i m a l charge 6dA. The integral i s taken over the entire surface of the s h e l l . Furthermore, since magnetic effects are neglected, the f i e l d i s written: F ( X 0 ) — V<t>(X0) 2.2-2 Here l f ( x " 0 ) i s the e l e c t r i c f i e l d at X*0 and V i s the three dimensional gradient operator. Therefore: 2.2-3 Next, assume current can only flow on the s h e l l surface. This assumption means there are no avalanche - 16 -breakdowns or gas discharges inside the s h e l l . Furthermore assume the s h e l l has isotropic and Ohmic conduction. Let the conductivity be Q, where Q i s a surface conductivity ( i . e . Ohms per square). The e l e c t r i c f i e l d tangent to the shel l ' s surface w i l l drive a surface current on the s h e l l : Ji-OEj. 2.2-4 where J± i s the surface current density on the s h e l l and tx i s the e l e c t r i c f i e l d tangent to the bulb's surface ( i . e . \ is current per length). The surface current must obey the continuity equation, therefore: Here the Vx operator i s the appropriate two dimensional operator, since the current i s constrained to the s h e l l ' s surface which i s two dimensional ( i . e . any position on the s h e l l requires the s p e c i f i c a t i o n of two coordinates). Substitute equation 2.2-4 into equation 2.2-5, to obtain: VA • flEA +|4 "° 2.2-6 To progress further, the properties of orthogonal c u r v i l i n e a r coordinates must be reviewed. F i r s t l y , the three dimensional gradient operator i s written: 3 ... i i 11 + i * ! I + h*l 2 _7 V*" hi 71/ h23C2 S3 U3 2 , 2 7 Here ^ , fc2 and £3 are generalized coordinates, a,  a 2 and &3 are unit vectors and h, , h 2 and h 3 are scaling factors which depend on the coordinate system used. The two dimensional divergence operator i s w r i t t e n : 3 7 • —L_ x h,h 2 2.2-8 Substitute equation 2.2-7 into equation 2.2-8 to obtain: Vx • Vx1-1 [ i _ ( h 2 i l \ + i _ / h 1 2.2-9 Let: Vi v i v v. h, h : 2.2-10 Substitute equation 2.2-3 into equation 2.2-6 to obtain: 2.2-11 Substitute equation 2.2-10 into equation 2.2-11 to obtain: -nvA2 <j> 6dA -ntf$ E+ | | -o 2.2-12 Equation 2.2-12 i s a general equation which describes the motion of charges, under an external f i e l d -V<$, on any s h e l l which can be described by holding one coordinate constant in some orthogonal c u r v i l i n e a r coordinate system. F i r s t l y I w i l l find the response of a s h e l l to a uniform s t a t i c e l e c t r i c f i e l d which i s suddenly applied on a 3 Morse,P.M. and Feshbach,H., Methods of Theoretical Physics ,McGraw-Hill Book Company,New York(1953T - 18 -sh e l l which i n i t i a l l y had no surface charge density. This w i l l be c a l l e d the "switch on" problem. Once the "switch on" problem has been solved i t i s easy to use these results to obtain the response of a s h e l l to an alternating e l e c t r i c f i e l d . The s t a t i c f i e l d induces a surface charge density, 6 , on the s h e l l . Eventually the f i e l d produced by the induced charge w i l l cancel the f i e l d i nternal to the s h e l l . Assume the surface charge density approaches equilibrium monotonically. This i s v a l i d since magnetic ef f e c t s are assumed to be ne g l i g i b l e . The response of the s h e l l w i l l be similar to an RC c i r c u i t . Hence: 6(r,t)«]TcHr)x(l-exp(-Xt:)) 2.2-13 X This i s v a l i d since any monotonic function can be written as the sum of exponentials. The external f i e l d i s applied at t=0. The X are the time constants of the response of the sh e l l and the 5 ( r ) ^ are the amplitudes of each response. These are the quantities which must . be determined. At equilibrium (t-*» ) equation 2.2-13 becomes: « ( r , - ) - ^ « ( r ) x 2.2-14 X This puts a constraint on the 5 ( r ) ^ s , since at equilibrium the f i e l d created by the induced charge density, 6(r,«»), must exactly cancel the applied f i e l d inside the s h e l l . This w i l l be c a l l e d the "equilibrium condition". It turns out the X's and the 6(r)j^s are the solutions of an eigenvalue problem, where the X*s are eigenvalues and - 19 -the 6 ( r ) x s are eigensolutions. It also becomes evident that the X's and 6(r)^s depend on the geometry of the s h e l l and the orientation of the e l e c t r i c f i e l d with respect to the s h e l l . The components of 6 ( r ) x w i l l be made up of a linear combination of the solutions to Laplace's equation in the coordinate system the problem i s posed. Furthermore they must be consistent with the "equilibrium condition". There i s interest in tubular bulbs since they possess axi-symmetric s e n s i t i v i t y . E l e c t r i c f i e l d monitors require t h i s sort of s e n s i t i v i t y for some applications. Hence the "switch on" problem w i l l be solved for a spheroidal s h e l l described by prolate spheroidal coordinates, since this s h e l l can be taken to a tubular or spherical l i m i t . It turns out t h i s problem can be solved a n a l y t i c a l l y for these l i m i t s . Note, surfaces w i l l always be chosen which are described by holding one coordinate constant in any chosen coordinate system since integrals l i k e the one appearing in equation 2.2-12 must be solved. With surfaces chosen in this manner 1/R appearing in the integrals w i l l simply be the appropriate Green's function for that coordinate system. This allows the integrals to be evaluated. If the s h e l l i s allowed to be of a r b i t r a r y shape i t w i l l be necessary to sum over an i n f i n i t e number of Green's functions to obtain the appropriate 1/R. This makes the problem hopelessly complicated. F i r s t l y the spheroid w i l l be long (asymptotically) such that i t i s similar to a c y l i n d e r . A long spheroid w i l l be c a l l e d a cylinder in the following analysis although i t w i l l - 20 -become apparent l a t e r there are s l i g h t differences in the responses of these two geometries. If the cylinder i s subject to the conditions s p e c i f i e d in the "switch on" problem i t i s clear that the response of the cylinder depends on i t s orientation with respect to the e l e c t r i c f i e l d . However one knows from e l e c t r o s t a t i c s that i f th i s problem i s solved for two orientations, then i t i s easy to extend these results to a l l orientations. This problem w i l l be solved with the e l e c t r i c f i e l d p a r a l l e l to the axis of the cylinder and perpendicular to the axis. These results w i l l be asymptotic since the spheroid w i l l be made long in an asymptotic manner. The results for the problem with the e l e c t r i c f i e l d perpendicular to the axis of the cylinder w i l l be checked by an exact c a l c u l a t i o n , but the res u l t s for the problem with the e l e c t r i c f i e l d p a r a l l e l to the axis are not readily soluble by any other method. Secondly the spheroid w i l l be taken to the spherical l i m i t (asymptotically) and the "switch on" problem solved. Clearly the results w i l l be independent of the orientation of the sphere with respect to the f i e l d . These results w i l l be checked by an exact c a l c u l a t i o n . To obtain the eigenvalue problem, substitute equation 2.2-13 into equation 2.2-12 to obtain: V 6 ( l - e x p ( - X t ) ) d A ^ * Air e 0 R n » ^ + L * « x « P M t ) - 0 2.2-15 A Assume the time constants, X , are independent of sp a t i a l coordinates. Hence the decay rates can be equated to obtain: - 21 -7l<h X j rr + V.'cJ) -0 E 2.2-16 and: /M A + X 6 X - ° 2.2-17 Equation 2.2-16 i s a statement of the "equilibrium condition". To demonstrate t h i s substitute equation 2.2-14 into equation 2.2-16 to obtain: _ 2 i 6 (r,»)dA . „ 2- _ n V * 9 <We0R + V ^ E -0 2.2-18 Equation 2.2-18 may be written: 2.2-19 Inside the s h e l l the quantity between the brackets i s zero,, by. the "equilibrium condition", hence no new information can be gained from equation 2.2-16. To fi n d the response of a s h e l l due to the conditions s p e c i f i e d in the "switch on" problem one must solve equation 2.2-17. This equation i s an eigenvalue problem, where X are the eigenvalues and a r e t n e eigensolutions. Solving equation 2.2-17 w i l l determine the eigenvalues exactly and the eigensolutions to within undetermined c o e f f i c i e n t s . Use of the "equilibrium condition" w i l l remove a l l undetermined - 22 -coef f ic ients. 2.3 THEORY PARTICULARIZED TO PROLATE SPHEROIDAL COORDINATES To obtain the eigenvalue problem in terms of prolate spheroidal coordinates V 2, 1/R, fi(*)A and dA, must be found in terms of these coordinates. Prolate spheroidal coordinates are obtained by rotating e l l i p t i c a l coordinates about the major axis of the e l l i p s e s . Suppose the foci of the e l l i p s e s are located at ±a/2 on the Z-axis as shown in figure 7. Let r1 be the distance between an a r b i t r a r y point P and +a/2 on the Z-axis and l e t r2 be the distance between point P and -a/2 on the Z-axis. Define £ and n to be: r. f i + r 2 2.3-1 * a r, -r2 a 2.3-2 where "a" i s the distance between the f o c i . Rotate t h i s coordinate system through an angle i about the Z-axis as shown in figure 8. • i s the angle between the plane formed by r1 and r2 and the X-axis. Thus, the position of any a r b i t r a r y point P can be described by specifying r,, n and i. This i s the prolate spheroidal coordinate system. From the d e f i n i t i o n s of £, n and 4 i t i s clear that £ must be between 1 and », n must be between 1 and -1, and i must be between 0 and 2*. Furthermore i t i s clear from equation 2.3-1 that a constant £=£ 0 w i l l describe a spheroidal s h e l l . This i s the - 23 -ELLIPTICAL COORDINATES — z FIGURE 7 PROLATE SPHEROIDAL COORDINATES P -—T^ i N Y \ FIGURE 8 - 24 -geometry desired for th i s problem. Any point on th i s s h e l l i s described by the correct s p e c i f i c a t i o n of n and 4. Moreover, rectangular coordinates x,y, and z can be obtained in terms of £,n and 4. The results are: 3 x- -ja/( r / - l ) (1-n*) Cos$ 2.3-3 y - j a / ( ^ - 1 ) (l-n«) S i n * 2.3-4 8 " i ' 5 n 2.3-5 The scaling factors which enter equation 2.2-10 are found by equating the length of an i n f i n i t e s i m a l arc, dS, between two coordinates systems, as shown below: 3 dS 2«dx 2+dy 2+dz 2«h 2dS 2+h 2dn 2+h 2d* 2 ' 2.3-6 If variables n and 4 are not allowed to vary then equation 2.3-6 becomes: 2.3-7 Substitute equations 2.3-3, 2.3-4, and 2.3-5 into equation 2.3-7 to obtain: h r 2 a \T^T~ 2.3-8 S i m i l a r l y : - 25 -h n " '* " 2.3-9 h ^ a / ( l - n ' ) ( ? ' - l ) 2.3-10 The variables which enter the operator V 2 are n and 4 since a spheroidal surface i s defined by constant £=£ 0. Thus, substitute equations 2.3-9 and 2.3-10 into equation 2.2-10 to obtain: • ' ^ t ' - n * a n / t ' - n * an + <^- i )< i- n * ) a~F/ 2 - 3 " 1 1 The area element, dA, in equation 2.2-16 i s written: dA- h dn d * 2.3-12 where the scale factors h and h a enter equation 2.3-12 in order to conserve arc lengths ( i . e . h^dn * s a length whereas dn i s not). Substitute equations 2.3-9 and 2.3-10 into equation 2.3-12 to obtain: d A ^ 2 ( C 2 - n 2 ) J s ( C 2 - D l s d n d * 2.3-13 An appropriate representation of 1/R i s the Green's function for an i n f i n i t e domain in terms of prolate spheroidal coordinates. The Green's function i s the solution of the equation shown below solved in an i n f i n i t e domain: - 26 -V 2 G ( X , X 0 ) - - 4 T ? 6 3 ( X - X 0 ) 2 . 3 - 1 4 where x0 i s an arb i t r a r y fixed point in three dimensional space and x i s a variable point. The Green's function, G, i s a function of both x0 and x. The operator V 2 i s the three dimensional Laplacian only involving the derivatives of x. The solution of equation 2 . 3 - 1 4 can be obtained with the use of Fourier transforms. The resu l t i s : G = Ix-x-J " I 2 . 3 - 1 5 where the v e r t i c a l bars mean the magnitude of the quantity between them i s to be taken. Hence R i s the distance between xD and x. The Green's function for an i n f i n i t e domain in terms of prolate spheroidal coordinates i s solved in Morse and Feshbach . The result i s : 3 where e m i s the Neumann factor which i s equal to 1 for m=0 and equal to 2 for m>0 . P j J ( n ) i s a Legendre function of the f i r s t kind and Q™(0 i s a Legendre function of the second kind and i i s the square root of - 1 . Furthermore £0, n 0 and • 0 are the coordinates of the fixed point and £,n and i are the coordinates of the variable point. The quantity R in equation 2 . 2 - 1 7 i s the distance between two points on a - 27 -spheroidal s h e l l e0 . Hence, set 0= £ 0 in equation 2.3-16 to obtain: i - | £ ( 2 o + i ) 2 . B i j | 5 ^ 1 c . . [ - < « - d p ; < , 0 > F ; ( n ) F ; ( t 0 ) Q ^ t 0 ) n s O m s O ' L J L J 2.3-17 Up to th i s point a l l the quantities necessary to solve the eigenvalue problem, equation 2.2-17, have been obtained except for selecting an appropriate form of 6 ( r ) x . To obtain insight into the appropriate form consider equation 2.2-17: /6\ dA 7nn-R + xV 0 2 . 2 - 1 7 Let: • • f t dA e 0R 2 . 3 - 1 8 Substitute equation 2.3-18 into 2.2-17 to obtain: Vjl + X« x -0 2.3-19 Substitute equations 2.3-17 and 2.3-13 into equation 2.3-18 to obtain: 6, ( t 2 - n 2 ) ! 5 a ( C 2 - l ) , i -u nsO m:o L J 0 -1 *Cos • ( • - • 0 ) P ; ( n ( , ) p ; ( n ) p ; ( C o ) Q ; ( C o ) 2 - 3 - 2 0 6 X must depend on n and 4 where n ranges from -1 to 1 and 4 - 28 -from 0 to 2*. However, p ^ n ^ f ° r m s a complete basis on L 2 ( - l , l ) and a linear combination of Cosv <and Sinv <forms a complete basis on L 2(0,2ir), where v i s an integer. This means any reasonable function of n and • can be written as a linear combination of these functions where the ranges of r\ and 4 are -1 to 1 and 0 to 2ir respectively. A reasonable function i s a bounded function which i s continuous and has a continuous f i r s t d e r i v a t i v e . Hence: 6X Uo^ 2) > S-2]R^P;(n)(co8v4 + C ^ S i n v ^ 2 . 3 _ 2 1 M v where Rv, and C v. are undetermined c o e f f i c i e n t s . This i s a y X y X convenient form of 6^  ( t^-n 2)" 4 because, i f equation 2.3-21 was substituted into equation 2.3-20 the integrals in " I " could be done a n a l y t i c a l l y since the integrand would contain only orthogonal polynomials. However, before t h i s i s done equation 2.3-21 can be s i m p l i f i e d considerably with the symmetry properties and equilibrium conditions of the problems that are to be solve. Consider the problem with the applied e l e c t r i c f i e l d in the Z-direction. In t h i s case the f i e l d i s p a r a l l e l to the axis of the spheroid. At equilibrium 6 ( r ) . can have no i A dependence due to the symmetry of the equilibrium charge density d i s t r i b u t i o n , hence v=0 . Furthermore one can obtain an exact expression for 6 2(r,»), where:3 e 0 E « p j ( n ) ^ 6z<'-->' 2h e Q yu 0 mi i - i> " V ( r ) ^ 2-3"22 A - 29 -Here 6 ( r , « ° ) i s the equilibrium charge density d i s t r i b u t i o n z r e s u l t i n g from a uniform applied f i e l d in the Z-direction and E i s the strength of the f i e l d . The second term in th i s equation was obtained from equation 2.2-14. 6(r). are the A Z eigensolut ions of the Z-direction problem ( i . e . *(r*)A2 s a t i s f i e s equation 2.2-17). The equilibrium charge density d i s t r i b u t i o n resulting from a uniform applied f i e l d in the X-direction has also been c a l c u l a t e d : 3 £ 0 E X 8 P ) ( n ) C o s i . _ ^ 6 x ( r ' - > " 2h Q { ( € 0 ) ( t J - D "2/ ( r )*x 2.3-23 A 6 x (r , «0 i s the equilibrium charge density d i s t r i b u t i o n re s u l t i n g from a uniform e l e c t r i c f i e l d , E x, applied in the X-direction. The second term in t h i s equation was obtained from equation 2.2-14. *(r)^ x are the eigensolutions for the X-direction problem ( i . e . 6 ( r ) . s a t i s f i e s equation 2.2-17). Hence, v=l in th i s case. Thus, from an inspection of 2.3-22 and 2.3-23 equation 2.3-21 becomes: «(r) A z(C 0 2-n 2) i $-^R p° xP;(n) 2.3-24 and: 6(')XxU?"n2)l5"SRJxPt(n)Co8* • 2.3-25 M The exact equilibrium charge density d i s t r i b u t i o n s , 6 ( r , » l and 6 ( r , « ) , are not used in t h i s analysis, but Z X rather the expansions of S ( r ) X z ( £ 2 - n 2 ) J i and 6 ( r ) X z U j - n 2 ) 1 * in terms of orthogonal polynomials because, the exact - 30 -equilibrium d i s t r i b u t i o n s are not eigensolutions, hence they do not s a t i s f y equation 2.2-17. Substitute equation 2.3-24 into equation 2.3-20 and evaluate " I " . The result i s : where 1^ i s the value of " I " r e s u l t i n g from an applied f i e l d in the Z-direction. Substitute equation 2.3-25 into 2.3-20 and evaluate " I " . The result i s : where I x i s the value of " I " r e s u l t i n g from an applied f i e l d in the X-direction. To obtain the eigenvalue problem res u l t i n g from an applied f i e l d in the Z-direction consider equation 2.3-19: V?I + X« x -0 • 2.3-19 Substitute equations 2.3-26, 2.3-24 and 2.3-11 into equation 2.3-19, then take the moment of the resulting equation with respect to P°(n). Taking the moment of an equation with respect to a function means the equation i s multi p l i e d by the function and the entire equation integrated over the range of the variables for which the function i s defined. See appendix A for the detailed c a l c u l a t i o n . The result i s : LR° B° + X R° -0 2 3-28 u where: - 31 -ac 1 2.3-29 Here X z i s the eigenvalue r e s u l t i n g from an applied f i e l d in the Z-direction. Let B z be the tensor B° N written in matrix form, then the X's can be found with the equation: d e t [ B z * X z u ] = 0 2.3-30 where u i s the unit tensor. The eigenvectors R ° X can be found, to within undetermined c o e f f i c i e n t s , by solving: [ Bz * Xz UK°X °° 2 ' 3 " 3 1 To obtain the eigenvalue problem re s u l t i n g from an applied f i e l d in the X-direction substitute equations 2.3-27, 2.3-25 and 2.3-11 into equation 2.3-19, then take the moment of the resu l t i n g equation with respect to P*("n). See appendix A for the detailed c a l c u l a t i o n . The.result i s : ZR 1. B1 + X R* -0 2.3-32 yX M N x NX U where: B l . i 0 ^ - 1 > l ' < 2 W ^ ) p l f g ) o l f n / p l f n x < t MN a c 0 p (y + l)N(N+l) Pp ( 9 Q y ( V / P N ( n ) * -1 (,}-n2) i _ _ - L lEaL_ V < n ) d n 2.3-33 Here X x i s the eigenvalue r e s u l t i n g from an applied f i e l d in - 32 -the X-direction. Let B be the matrix form of tensor B 1 . Thus the A^s may be obtained by solving: d e t [ B x * X x u ] = 0 2.3-34 and the R*x may be obtained, to within undetermined c o e f f i c i e n t s , by solving: [ Bx + X X U ] R J A - ° 2 ' 3 " 3 5 It i s clear from equations 2.3-29 and 2.3-33 that the matrices B are i n f i n i t e in dimension. However, for now suppose the dimension of the matrix i s chosen to be f i n i t e . Furthermore suppose a l l the eigenvalues, A , and corresponding eigenvectors, R^x , are calculated ( i . e . solve equations 2.3-30 and 2.3-34 respectively, then use these res u l t s in equations 2.3-31 and 2.3-35 to obtain the the R^x to within undetermined c o e f f i c i e n t s ) . Next the eigensolutions, > c a n ke found to within these undetermined c o e f f i c i e n t s with equations 2.3-24 and 2.3-25 respectively. F i n a l l y the eigensolutions, 6 ^ ( r ) , are summed so that they add up to the equilibrium charge density d i s t r i b u t i o n , 6(r,«), where: 6(r . - ) - ^ * ( r ) x 2.2-14 A A l l undetermined c o e f f i c i e n t s are removed with equation 2.2-14. It turns out the "equilibrium condition" can be s a t i s f i e d regardless of the dimension of the matrix B. Hence the response of the s h e l l i s : - 33 -6 ( r , t ) - ^ 6 ( r ) A ( l - exp ( -X t ) ) 2.2-13 6 N X ( r ) i s used to denote the eigensolutions obtained by solving the N-dimensional approximation to the i n f i n i t e dimensional matrix. This i s obtained by setting a^=0 for i or j > N where a^. i s an element of the matrix. Furthermore, when t h i s approximation i s made the response of the s h e l l i s denoted by * N ( ' » t ) - As the dimension of the matrix increases, i t turns out that 6 N ( r , t ) rapidly converges such that 6 N ( r , t ) i s e s s e n t i a l l y indistinguishable from 4 N + 1 ( r , t ) , as expected. Furthermore, i t turns out that the convergence is so rapid, in our examples, that 6 7 ( r , t ) can be used as an approximation to 6n(.r,t). 2.4 THE CYLINDRICAL LIMIT OF A SPHEROID The response of a spheroidal s h e l l under the influence of an e l e c t r i c f i e l d applied p a r a l l e l or perpendicular to i t s axis can be calculated with the information obtained thus f a r . There i s only one problem, the integrals in equations 2.3-29 and 2.3-33 cannot be done a n a l y t i c a l l y , they must be done numerically. To avoid t h i s , two asymptotic l i m i t s of a spheroid w i l l be considered. F i r s t l y , the spheroid w i l l be made long (asymptotically) so i t i s l i k e a cyl i n d e r . Secondly, the spheroid w i l l be allowed to approach a sphere (asymptotically). It turns out the integrals can be done a n a l y t i c a l l y in these l i m i t s . - 34 -In the f i r s t l i m i t l e t the r a t i o of the length to diameter of the spheroid become a r b i t r a r i l y large. Consider equations 2.3-3 and 2.3-4. It i s clear, for a spheroidal s h e l l £=e;0, that X and Y take on the i r maximum values at n-0 (the centre of the spheroid). Thus: X M " " J * * T O - 1 COB* 2.4-1 Y M " J a / S o " 1 S i n * 2.4-2 where X M and Y M are the maximum values of X and Y respectively. The maximum radius of the spheroid i s : ^-/Xg+Y* - ya/^FT 2.4-3 As the spheroid becomes long, i t w i l l become c y l i n d r i c a l in appearance. The diameter of the cylinder (long spheroid) w i l l be defined to be: D-2r M 2.4-4 Moreover, i t i s clear from equations 2.3-3 and 2.3-4 that X and Y are zero at thus from equation 2.3-5 i t can be seen that the maximum value of Z i s : 2 M - -jaCo 2.4-5 Hence, the length of the cylinder (long spheroid) i s : - 35 -L « 2 Z M « a e 0 2.4-6 Therefore: £ * / # T 2.4-7 L/D i s c a l l e d the aspect r a t i o of the cylin d e r . Furthermore, from equation 2 . 4 - 7 , as £ 0 approaches one L/D approaches i n f i n i t y ( i . e . the spheroid becomes c y l i n d r i c a l ) . Let: ^ o " 1 + A 2.4-8 where A i s small but greater than zero. Substitute equation 2.4-8 into equation 2.4-7 to obtain: L 1+A » " <2«*(l* | >* 2 - 4 " 9 Equation 2 . 4 - 9 can be series expanded to obtain: 5 " 7 T T % ( l + 7A + O U 2 ) > 2 . 4 - 1 0 (2A) It i s clear from equation 2 . 4 - 1 0 that as A approaches zero L/D approaches 1/(2A)"* . In the following analysis I w i l l only keep the f i r s t term of any such series. Thus: L ^ 1 D k 2.4-11 D (2A>* where the ^ means the right hand quantity approaches the l e f t hand quantity asymptotically as A becomes small. Furthermore: - 36 -r *u al r 1 ' ' 2.4-12 A i s the parameter which must be small for the asymptotic expansions to be v a l i d . A< 0.1 w i l l be considered small. Hence, from equation 2.4-11, L/D>3 i s the range of v a l i d i t y of a l l the asymptotic expansions. To obtain the asymptotic eigenvalue equations for the "switch on" problem with the e l e c t r i c f i e l d applied p a r a l l e l to the axis of the cylinder substitute equation 2.4-8 into equation 2.3-29 and keep only zero order terms in A to obtain: > , R°. B° + A R° - 0 fcjj-' pA pN z NA 2.3-28 B M 1 "1 2.4-13 The equilibrium charge density d i s t r i b u t i o n for the Z-direction problem, equation 2.3-22, becomes: « 0 E 8 pO-<n> -6 ( r , - ) ~ 2 ' (l - n 2) l sQ°(l+ A)(2A) l s - 2.4-14 and the eigensolutions, equation 2.3-24, become: ZR° P° ( n ) . , _ „ 2 x i s 2.4-15 To obtain the asymptotic eigenvalue equations for the "switch on" problem with the e l e c t r i c f i e l d applied - 37 -perpendicular to the axis of the cylinder substitute equation 2.4-8 into equation 2.3-33 and keep only the largest terms to obtain: ZR ' + )i R ' *0 O pX U N x NX 2.3-32 U t in<2N+i)Qj(l+A)pi(l+A) f?l<n)P*(n)dn 2 4-16 J ( l - n 2 ) 1 * Li - — -W N ac 0y (u + l)N(N+l) (2A) 4 The equilibrium charge density d i s t r i b u t i o n for the X-direction problem, equation 2.3-23, becomes: «x(r,«) % e 0 E x C o s * / l ^ n ^ 2.4-17 The eigensolutions, equation 2.3-25, become: *»»(r> y > (1.n.,^  2-4-,e The integrals in 2.4-13 and 2.4-16 can be solved a n a l y t i c a l l y . 2.5 APPROXIMATE SOLUTION OF THE "SWITCH ON" PROBLEM WITH THE APPLIED FIELD PERPENDICULAR TO THE AXIS OF THE SPHEROID F i r s t l y the "switch on" problem for a long spheroid with the applied e l e c t r i c f i e l d perpendicular to i t s axis w i l l be solved. This i s done in appendix B. Here I w i l l b r i e f l y outline the procedure and give the r e s u l t s . F i r s t l y the dimension of matrix B i s chosen. Next, the integral in equation 2.4-16 i s performed. Then the eigenvalue problem, - 38 -equation 2.3-32, i s solved for the chosen matrix, say N-dimensional. The eigenvalues are obtained by solving: detCB 1 • X u)=0 2.3-34 and the eigenvectors, Rj^ , are found to within undetermined c o e f f i c i e n t s by solving: ( B J N * X x u ) R l x -° 2.3-35 Hence, the eigensolutions can be found to within undetermined c o e f f i c i e n t s with equation 2.4-18: Pj j (n)Cos4 " ( l _ n l ) % 2.5-1 Here the N in $ N A X ( r ) denotes the dimension of the matrix. F i n a l l y a l l undetermined c o e f f i c i e n t s can be removed with the use of the "equilibrium condition", equations 2.2-14, 2.4-17 and 2.5-1: ^ * 2.2-14 P l ( n > 2.4-17 Substitute equations 2.5-1 and 2.4-17 into equation 2.2-14 to obtain: £ o x . - - ^ M A " y 2.5-2 XM The undetermined c o e f f i c i e n t s in R j x are removed with equation 2.5-2. Hence the eigensolutions, 6 fr) , are NXX V ' - 39 -known. F i n a l l y the response of the s h e l l i s obtained with equation 2.2-13: 6 N X ( ? » ^ - X > N X x ( ? ) U - e * p ( - X t > ) 2 2 _ 1 3 X where the N in 6 (r,t) denotes the dimension of the matrix. This c a l c u l a t i o n is done in appendix B. The res u l t s are given below. Note, i t turns out only the elements of the matrix in which both y and N are odd need to be considered in the c a l c u l a t i o n , t h i s i s demonstrated in appendix B. The resu l t s of the ca l c u l a t i o n are: « l x ( ? . t ) « 0 «,c.M(i-«,(- ^ f £ a „ 2 5 _ 4 « 3 x ( ? . t ) * e 0 E Co.•(. 8349- . 34 73 (Sn 2 -1) ) ( l-«*p<- - " 8 Q t ) ) + t„E ] tCo « « ( . U 5 H - . 3 4 7 3 ( S n 2 - D>a-«xp(- •8"0t)> 2.5~5 c„ E xCo« •(.6 3 7 4 - . 4 1 4 0 ( 5 n 2 - l ) + .17*6(21n l , -14n 2 +l) ) (l-«xp(- ' " * 0 t ) ) + c 0 E Co . » ( . 3 1 8 2 + . 2 9 1 0 ( 5 t l 2 - l ) - . 3 2 1 6 ( 2 1 n , ' - U f i 2 + l ) ) ( l-«xp<- - 6 * 9 0 t ) ) + e 0 E x C o « 4 ( . 0 4 4 4 + . 1 2 3 0 ( 5 f l 2 - l > + . 1 4 7 2 ( 2 1 t i , , - l 4 t i 2 + l ) ) ( l - « x p ( - ' j . 2 * 0 ' ) ) t o 2 . 5~6 - 40 -« 7 X<r.t> + t 0 E x C o . » ( . 5 0 9 3 - . 3 7 53(5n2-J) + . J556(21nl,-l*n2+l)-.0209(429ns-*95n''+135n2-S)) ( l - « x p ( + c0 E x Co«» (.3408+. 0878 (5n2-l)-.3602 (2In*-Un2+1) + . 0472 (429n*-495nl,+ l35n 2 -5) ) ( l - « x p ( + i 0 E x C o » •(.1337+.238l(Sn2-D+.029*(21n''-Un2+l)-.0413(*29n'-*95ri , ,+135n*-5)) ( l - « x p ( + e0ExCo»f(.0162+.0495(5n2-D+.0476(2lTi , ,-I4D2+l)+.0151(429n*-495n''+135n2-S)) ( l - « x p ( These functions are plotted on the following pages; * 1 x/e 0E xCos< i s plotted against tfl/Re 0 in figure 9, 6 /e E Cos i in figure 10, 6 /e„E Cos* in figure 11, and 3 X o x ^ sx' 0 x T 3 ' 6 7 X/e 0E xCos< in figure 12, for various n . To demonstrate that these functions are converging as N increases, 6 3 X / e 0 E X C ° S * ' 6 5 X / E ° E X C O S * 3 1 1 6 6 7 X / E 0 E X C O S * P l o t t e d against tO./Re0 and overlayed for n = 0 and n=.75 in figure 13. From this figure one can see that these functions converge rapidly to fi^C^t). Later an i n f i n i t e cylinder "switch on" problem with the applied f i e l d perpendicular to i t s axis w i l l be solved exactly. This solution makes i t possible to predict the form of 6 e a X ( r , t ) for a long spheroid through the s i m i l a r i t y of the geometries. It turns out, as w i l l be shown l a t e r , «. x(r,t) i s : «. x(r,t) * e 0E CoB*Yl-exp( 0 t \\ 2.5-8 hence: Rc0 •liiOt)) 2.5-7 - 41 -FIGURE 9 - 42 -FIGURE 10 - 43 -THEORETICAL 6 ^ ( r , t ) / e 0 E ^ C o s j PLOTTED AGAINST nt/Re 0 FIGURE 11 - 44 -THEORETICAL g y ^ ( r , t ) / c 0 E y C o s 4 PLOTTED AGAINST flt/Re0 n s . 75 FIGURE 12 - 45 -DEMONSTRATION OF THE CONVERGENCE OF 6 ,6 AND 6 • — 3 X1 5X 7 X (A FIGURE 13 FIGURE 14 - 47 -A ^ x 2 e 0 R ( l - n 2 ) l s 2.5-9 The reason t h i s r e s u l t was not obtained by an exact solution of equations 2.2-16 and 2.2-17 i s that these equations were derived assuming A was independent of s p a t i a l coordinates. Hence, these equations could be re-derived taking into account the s p a t i a l derivatives of A and the problem solved exactly. However, th i s would complicate the problem considerably. Moreover, t h i s i s the only example considered which has a s p a t i a l l y dependent A. 6 /e„E Cos 4 and 6 /e DE Cos * are plotted in figure 14 « e x O X 7X ° X J for various n. The agreement between these functions i s excellent. 2.6 APPROXIMATE SOLUTION OF THE "SWITCH ON" PROBLEM WITH THE APPLIED PARALLEL TO THE AXIS OF THE SPHEROID The spheroid has been defined so that i t s axis i s p a r a l l e l to the Z-axis. Hence, in t h i s section the "switch on" problem for a long spheroid with the e l e c t r i c f i e l d applied in the Z-direction w i l l be solved. This c a l c u l a t i o n i s done in appendix C. Here I w i l l b r i e f l y outline the procedure used to solve t h i s problem and give the results of the c a l c u l a t i o n . F i r s t l y the dimension of the matrix B° i s chosen and a l l the elements found. The eigenvalues , A , are found by solving equation 2.3-30: - 48 -d e t ( B ° N • X zu)=0 2.3-30 and the eigenvectors, , are found to within undetermined c o e f f i c i e n t s with equation 2.3-31: ( B 2N + Az u ) R 2 x = 0 2.3-31 Hence the eigensolutions, 6 ( r ) , may be found to within these undetermined c o e f f i c i e n t s with equation 2.4-15: ZR° P°(n ) where N appearing in * N A Z ( r ) denotes the dimension of the matrix. The undetermined c o e f f i c i e n t s may be removed with the "equilibrium condition", equations 2.2-14, 2.4-14 and 2.6-1. Substitute equations 2.4-14 and 2.6-1 into 2.2-14 to obtain: uX y x " 2.6-2 A l l previously undetermined c o e f f i c i e n t s may be removed with equation 2.6-2. Therefore, the eigensolutions, * N A Z ( r ) , w i l l be known exactly. F i n a l l y the response of the s h e l l , 6 ( r , t ) , i s found with equation 2.2-13: N Z 6 N Z ( r . t ) - ^ « N A l ( ? ) ( l - e x p ( - A 8 t ) ) 2.2-13 A where N in 6 , ( r , t ) denotes the dimension of the matrix used N Z in the c a l c u l a t i o n . I would l i k e to obtain the quantity - 49 -5 ,(*»*)• I f c turns out 6 ( r , t ) converges very rapidly to ^ X NX 6 o o z ( r , t ) as the dimension of matrix B increases. Thus a reasonable approximation to 6 ( r , t ) i s obtained with a minimum amount of work. This c a l c u l a t i o n i s done in appendix C. The results are: « . 2 ( r , t ) % — — , - _ — ( l -exp( -X t) ) DQjf 1+ firW -n 2 )* z 2 ' 6 3 where: X z * - - H 1 J 2 - 2 6 2.6-4 where D i s the maximum diameter of the spheroid and L i s the length of the spheroid. Note, the time constant in t h i s case is an empirical f i t to the the o r e t i c a l time constant as shown in appendix C. 2.7 APPROXIMATE SOLUTION OF THE "SWITCH ON" PROBLEM IN THE SPHERICAL LIMIT WITH THE APPLIED FIELD IN THE X-DIRECTION A spheroidal s h e l l i s described in . prolate spheroidal coordinates by holding £ constant. This s h e l l i s taken to the spherical l i m i t by allowing 5 0 to approach i n f i n i t y as shown below. From equations 2.3-3, 2.3-4 and 2.3-5 i t i s easy to show: - 50 -x 2+y 2+z 2^ i a 2 U 2-l+ n 2 > 2 > ? _ 1 Let £0-^» . n varies between -1 and 1 hence, as £ 0 becomes large, the last two terms in equation 2.7-1 may be ignored. Thus: x 2+y 2+z 2~ ja2£2 where t 0>>l 2.7-2 Hence the radius of the sphere i s : R * J*S>o 2.7-3 In t h i s section the response of a spheroidal s h e l l to an e l e c t r i c f i e l d applied in the X-direction w i l l be found. The equations which govern th i s response are; 2.3-32, 2.3-33 and 2.3-25 with the "equilibrium condition", equation 2.3-23. Take these equations to the spherical l i m i t by allowing f,0 to approach i n f i n i t y to obtain: ERl B1 + X R1, -0 2.7-4 yX M N x NX i O(2N+ l ) P j ; ( E ) Q j ( Q .1 „ V o V o U N ac 0y (u + l)N(N+l) 1 2.7-5 o M 2.7-6 The equilibrium charge density d i s t r i b u t i o n becomes: - 51 -« x ( r , » ) * 3c 0E x P } ( n)Cos• 2.7-7 This problem is solved in appendix D. Here I w i l l b r i e f l y outline the procedure used to solve the problem and give the results of the c a l c u l a t i o n . F i r s t l y the integral in equation 2.7-5 i s performed, hence the B 1 are found. Next the eigenvalues, x , are found with: [ Bx * X x U ] = ( det| „ • „u|=0 2.3-34 and the eigenvectors, R1 , are found to within undetermined c o e f f i c i e n t s with: [Bx + Xx u]RJx "° 2.3-35 The undetermined c o e f f i c i e n t s are removed with equations 2.7-6, 2.7-7 and the "equilibrium condition", equation 2.2-14. Substitute equations 2.7-6 and 2.7-7 into equation 2.2-14 to obtain: 3'o * ] £ i R i x 2.7-8 AM Equation 2.7-8 i s used to remove a l l undetermined c o e f f i c i e n t s . Hence the response of the s h e l l can be found with equation 2.2-13: « ( r , t ) - ^ 6 ( r ) x ( l - exp ( -X t ) ) 2.2-13 The re s u l t s are: - 52 -« x ( r , t ) * 3 e 0 P j ( n)Cos « ( l - e x p ( - A t ) ) 2.7-9 where: A x 2n 3e 0 R 2.7-10 2.8 APPROXIMATE SOLUTION OF THE "SWITCH ON" PROBLEM IN THE SPHERICAL LIMIT WITH THE APPLIED FIELD IN THE Z-DIRECTION If the mathematical analysis i s correct, then in the spherical l i m i t the results should not depend on the orientation of the bulb with respect to the f i e l d . In t h i s section, as in the l a s t section, I l e t £ -*•<*> to obtain the spherical l i m i t of a spheroid. The equations which govern the response of a spheroidal s h e l l for the "switch on" problem with the f i e l d applied in the Z-direction are; 2.3-28, 2.3-29 and 2.3-24 with the "equilibrium condition", equation 2.3-22. In the l i m i t , C 0^ C D , these equations become: B° ^ n(2N+l)Q°U )P,?U) yi o u o P ° N ( n ) ^ ( l - r , 2 ) ^ P ) ; ( n ) d n 2.8-1 - 1 2.8-2 2.8-3 The equilibrium charge density d i s t r i b u t i o n becomes: - 53 -6 z * 3c oE zP0(n) 2.8-4 This problem i s solved in appendix E. Here I w i l l b r i e f l y outline the procedure used in t h i s c a l c u l a t i o n and present the re s u l t s obtained. F i r s t l y the integral in equation 2.8-1 i s performed, hence the B° are known. Next the eigenvalues, X , are JIN 3 z found with: det[B z • X z i i j = 0 2 > 3 _ 3 0 and the eigenvectors , , are found to within undetermined c o e f f i c i e n t s with: [Bz + Xzu]Rp°x "° 2.3-31 To remove the undetermined c o e f f i c i e n t s consider the "equilibrium condition". Substitute equations 2.8-3 and 2.8-4 into equation 2.2-14 to obtain: 3 e o E 2 P 0 ( n ) ^2\Kx P y ° ( n ) 2 ' 8 ~ 5 A M A l l undetermined c o e f f i c i e n t s are removed with equation 2.8-5. Thus the eigensolutions are known with equation 2.8-3. F i n a l l y the response of the s h e l l i s found with equation 2.2-13: «(r.t ) -^p«(r) A(l-exp(-Xt)> 2.2-13 The r e s u l t s are: - 54 -6 z ( r , t ) % 3 e 0 E z p O ( n ) ( l - e x p ( - X z t ) ) 2.8-6 where: z 2.8-7 As expected the solution for 6 (r,t) i s independent of the d i r e c t i o n of application of the applied f i e l d (see compared with the exact solution for a spherical bulb, which i s presented below. 2.9 EXACT SOLUTION OF THE "SWITCH ON" PROBLEM FOR A SPHERE WITH THE APPLIED FIELD IN THE Z-DIRECTION Consider a conducting sphere described by spherical coordinates as shown in figure 15. The response of this s h e l l to a uniform s t a t i c e l e c t r i c f i e l d applied at t=0 in the Z-direction w i l l be found. This can be accomplished with the use of equation 2.2-17: and with the use of the equilibrium charge density d i s t r i b u t i o n : ** in section 7). These approximate results are 2.2-17 6 z " 3 E O E Z C O 8 0 - 3E 0 E aPj (Co«e) 2.9-1 k Jackson,J.D., C l a s s i c a l Electrodynamics , John Wiley and Sons, New York(l9T5~l - 55 -SPHERICAL COORDINATES r Z ' y e / Y FIGURE 15 CYLINDRICAL COORDINATES z | FIGURE 16 - 56 -H e r e i s t h e m a g n i t u d e o f t h e e l e c t r i c f i e l d . I t t u r n s o u t , i n t h i s p r o b l e m , t h a t t h e r e i s o n l y one e i g e n s o l u t i o n , h e n c e t h e e i g e n s o l u t i o n i s e q u a l t o t h e e q u i l i b r i u m c h a r g e d e n s i t y d i s t r i b u t i o n . T h u s : 6z( ^ » - ) ' 6 Xz ( *> 2.9-2 To show t h i s a s s u m p t i o n i s c o r r e c t one c a n s u b s t i t u t e e q u a t i o n 2.9-2 i n t o e q u a t i o n 2.2-17 a n d d e m o n s t r a t e c o n s i s t e n c y . F u r t h e r m o r e t h e G r e e n ' s f u n c t i o n i n an i n f i n i t e d o m a i n i n t e r m s o f s p h e r i c a l c o o r d i n a t e s , w i t h r = r Q , i s : 3 G " * " E E 6 * T^fTT ' : < c o . V ' ; ( c o . e ) c o . ( . ( ? - * 0 ) ) I 2 > g _ 3 nso m=o 0 * where r 0 i s t h e r a d i u s o f t h e s p h e r e . M o r e o v e r dA i s : dA - r 2 S i n 0 d e d $ 2.9-4 an d V 2 i s : 1 * V 2 - 1 i -csinei^) + — i | i 2 2.9-5 S u b s t i t u t e e q u a t i o n s 2.9-2, 2.9-3, 2.9-4 a n d 2.9-5 i n t o e q u a t i o n 2.2-17 a n d p e r f o r m t h e r e s u l t i n g i n t e g r a t i o n . Then p e r f o r m t h e s p a t i a l d e r i v a t i v e s r e s u l t i n g f r o m V 2 a n d c a n c e l a s many t e r m s a s p o s s i b l e . T h i s c a l c u l a t i o n i s e a s y t o p e r f o r m . E q u a t i o n 2.9-2 t u r n s o u t t o be c o n s i s t e n t w i t h e q u a t i o n 2.2-17 . The r e s u l t o f t h e c a l c u l a t i o n i s : 6 z ( r , t ) - 3 e 0 E z C o s e ( l - e x p ( - X 2 t ) ) 2.9-6 w h e r e : - 57 -A* ' 3e0ro 2.9-7 The exact results for the sphere, equations 2.9-6 and 2.9-7, are consistent with the approximate results for a sphere, equations 2.8-6 and 2.8-7. 2.10 EXACT SOLUTION OF THE "SWITCH ON" PROBLEM FOR A CYLINDER WITH THE APPLIED FIELD PERPENDICULAR TO ITS AXIS Consider a conducting i n f i n i t e cylinder described by polar coordinates as shown in figure 16. The response of this s h e l l to a uniform s t a t i c e l e c t r i c f i e l d applied suddenly at t=0 in the X-direction w i l l be found. This can be accomplished with the use of equation 2.2-17: J dA n V ^ + X 6 A - ° 2.2-17 and the equilibrium charge density d i s t r i b u t i o n , 6 Cr,"), where: 6 x ( r , « 0 - c Q E x C o s 6 2 . 1 0 - 1 It turns out, in t h i s problem, that there is only one eigensolution, hence the eigensolution i s equal to the equilibrium charge density d i s t r i b u t i o n . Thus: 6x(r,-> " * X X < * > 2 . 1 0 - 2 To demonstrate t h i s assumption i s correct substitute - 58 e q u a t i o n 2.10-2 i n t o e q u a t i o n 2.2-17 and show t h e r e s u l t i s c o n s i s t e n t . F u r t h e r m o r e t h e G r e e n ' s f u n c t i o n i n an i n f i n i t e d o m a i n i n t e r m s o f c y l i n d r i c a l c o o r d i n a t e s , w i t h r = r 0 , i s : 3 03 G- £ - -21n [ r J+2y^Cos(n(»-» 0 ) ) 2.10-3 where r 0 i s t h e r a d i u s o f t h e c y l i n d e r . M o r e o v e r dA i s : dA « r 0 d 6 2.10-4 a n d V 2 i s : V? " rT 3~62 2.10-5 1 3 2 o S u b s t i t u t e e q u a t i o n s 2.10-2, 2.10-3, 2.10-4 a n d 2.10-5 i n t o e q u a t i o n 2.2-17 a n d p e r f o r m t h e r e s u l t i n g i n t e g r a t i o n . Then p e r f o r m t h e d e r i v a t i v e s r e s u l t i n g f r o m V 2 a n d c a n c e l a s many t e r m s a s p o s s i b l e . T h i s c a l c u l a t i o n i s e a s y t o p e r f o r m . E q u a t i o n 2.10-2 t u r n s o u t t o be c o n s i s t e n t w i t h e q u a t i o n 2.2-17 . The r e s u l t o f t h e c a l c u l a t i o n i s : « v ( r , t ) - e. E C o s 6 ( l - e x p ( - X t ) ) 2.10-6 X u X * w h e r e : X 2 c ° r ° 2.10-7 The t i m e c o n s t a n t o f a i n f i n i t e c y l i n d e r , A ^ , d e p e n d s on l/r0. A l o n g s p h e r o i d c o u l d be t h o u g h t o f b e i n g s i m i l a r t o a c y l i n d e r , e x c e p t i t s r a d i u s i s a f u n c t i o n o f s p a t i a l c o o r d i n a t e s . The r a d i u s o f a l o n g s p h e r o i d d e f i n e d i n - 59 -DEFINITION OF THE RADIUS OF A SPHEROID r r • R/l-n z \ 1 1 1 FIGURE 17 z = 7* to" prolate spheroidal coordinates i s : r - R/l-n*. 2.10-8 as shown in figure 17. Here R i s the maximum radius of the spheroid. Hence the time constant for a long spheroid, X x , would be expected to be: ft x 2 e 0 R ( l - n 2 ) l s 2.10-9 Thus, the response of the s h e l l would be: « x ( r , t ) % e 0 E x C o 8 9 ( l - e x P ( - X x t ) ) 2.10-10 This i s in excellent agreement with the res u l t s of section - 60 -2.11 RESPONSE OF A CONDUCTING SHELL TO AN ALTERNATING ELECTRIC FIELD Let the potential which describes the f i e l d external to the bulb be $>E. Furthermore l e t : $ E ( r , t ) - * E ( r ) e x p ( i w t ) 2.11-1 Here u i s the frequency of the alternating e l e c t r i c f i e l d and <tE(r) is a potential which describes a uniform f i e l d . Hence as u approaches zero I obtain the same external potential as in the "switch on" problem discussed e a r l i e r . This problem w i l l be set up such that as u approaches zero the "switch on" problem i s obtained. Let the surface charge density induced by the external f i e l d be: " «4 E(r)exp(iut) 2.11-2 The induced charge density i s assumed to fluctuate at the same frequency as the external f i e l d . 6 E ( r ) i s a charge density d i s t r i b u t i o n which i s a solution to the "switch on" problem. Thus as u approaches zero a must approach 1 for thi s problem to approach the "switch on" problem. Furthermore a i s expected to approach 0 as u approaches • since the bulb w i l l not respond to a f i e l d of i n f i n i t e frequency. Substitute equations 2.11-1 and 2.11-2 into equation 2.2-12. u i s assumed to be small so magnetic e f f e c t s can be neglected, hence equation 2.2-12 i s v a l i d . The result i s : - 61 --flaV 2 <j> 4 C w e R - 07 2 * E* iau)6 E»0 2.11-3 Substitute equations 2.2-16 and 2.2-17 into equation 2.11-3. The summation over A in equation 2.2-16 has been dropped, since the solutions to the "switch on" problems, that have been solved, contain one eigenvalue, X. The result of the substitution i s : 6 E dA o _ 1_ , , <w 2.11-4 1 *X Notice that a approaches 1 as t o approaches 0 and a approaches 0 as u approaches i n f i n i t y as expected. Substitute equation 2.11-4 into 2.11-2 to obtain: { w ( r » t ) e x p ( l u t ) 2.11-5 1 + i r 6E (r) induces an e l e c t r i c f i e l d inside the s h e l l equal in. magnitude to the external f i e l d but opposite in d i r e c t i o n . Hence: l + *r - 2.11-6 Here £ A i s the applied e l e c t r i c f i e l d and tu i s the component of the e l e c t r i c f i e l d produced inside the s h e l l due to the surface charges. Note, as u tends to 0 the induced f i e l d , E*N , i s equal in magnitude but opposite in - 62 -di r e c t i o n to the applied f i e l d as expected. This i s consistent with the "switch on" problem. The e l e c t r i c f i e l d internal to the bulb, ? , i s : E. + Ek 2.11-7 Substitute equation 2.11-6 into 2.11-7 to obtain: 1 - i -Hi L 2.11-8 Thus the magnitude of the s h e l l ' s internal f i e l d i s : E. E. = 2.11-9 where E, i s the magnitude of the internal f i e l d and E A i s the magnitude of the applied f i e l d . Furthermore, from equation 2.11-8, the phase of the internal e l e c t r i c f i e l d with respect to the external e l e c t r i c f i e l d i s : 6 - T a n 1 — 2.11-10 u where e i s defined in figure 18. Now enough information has been obtained to predict the effe c t of " s t a t i c screening" on a c a l i b r a t i o n curve of a bulb. As a reminder, the " s t a t i c screening" model i s v a l i d below and up to threshold when applied to these bulbs. Consider equation 2.1-4: - 63 -DEFINITION OF PHASE SHIFT E, , INTERNAL F I E L D E A , A P P L I E D F I E L D FIGURE 18 T h i s e q u a t i o n i s v a l i d f o r b u l b s w i t h a n o n c o n d u c t i n g s h e l l . The " i n t " w i l l be d r o p p e d s i n c e , a s d e s c r i b e d e a r l i e r , t h e c a l i b r a t i o n c u r v e s a r e a c t u a l l y q u i t e l i n e a r , h e n c e : f - I * " 6 EQ * 2.11-11 I n t h e d e r i v a t i o n o f 2.11-11 i t was assumed t h e e l e c t r i c f i e l d i n s i d e t h e b u l b was u n a f f e c t e d by t h e s h e l l . To a c c o u n t f o r t h e f i n i t e c o n d u c t i v i t y o f t h e s h e l l s i m p l y r e p l a c e t h e a p p l i e d f i e l d , E A , by t h e i n t e r n a l f i e l d , E,. Hence e q u a t i o n 2.11-11 becomes: E. ui 2E A f A «E„ 1 + 2.11-12 H e r e f A i s t h e f r e q u e n c y o f t h e a p p l i e d f i e l d . T h i s e q u a t i o n d e s c r i b e s t h e " s t a t i c s c r e e n i n g c u r v e * . The s l o p e o f t h e - 64 -"s t a t i c screening curve" i s reduced with respect to the "zero conductivity curve" as expected. This equation explains the curve shown in figure 6. The reduction in the slope of the " s t a t i c screening curve" with respect to the "zero conductivity curve" only depends on X/u. X depends on the geometry, size and conductivity of the s h e l l . I have found X for a sphere and for a spheroid, where two X's are required to describe the response of a spheroid; one for the f i e l d p a r a l l e l to the axis of the spheroid and one for the f i e l d perpendicular to the a x i s . Furthermore t h i s theory predicts a phase s h i f t of the internal e l e c t r i c f i e l d with respect to the external f i e l d . This model i s v a l i d at threshold, therefore i t predicts the phase s h i f t of the pulses at threshold. 2.12 "RESET SCREENING" MODEL At high e l e c t r i c f i e l d s the " s t a t i c screening" model f a i l s to explain the pperation of the bulbs. This i s due to the avalanche breakdowns which occur at high f i e l d s . These avalanche breakdowns reset the e l e c t r i c f i e l d to 0 within the bulb and within the conducting material which makes up the surface of the bulb. The assumptions required to derive the "reset screening" model are i d e n t i c a l to the assumptions required for the " s t a t i c screening" model except, e x p l i c i t account i s made for the "resetting" of the e l e c t r i c f i e l d in the bulb af t e r each avalanche breakdown. These assumptions are; magnetic ef f e c t s can be neglected and the s h e l l - 65 -possesses isotropic and Ohmic conduction. To account for t h i s "resetting of the f i e l d " consider uniform external f i e l d which i s an i n f i n i t e ramp: Here "a" i s the parameter which determines the slope of the ramp, t i s time and z indicates the d i r e c t i o n of the f i e l d . Let t h i s applied f i e l d be created by the p o t e n t i a l : where tfr) creates a uniform f i e l d . Consider a bulb with no surface charge on i t s s h e l l at t=0. As the external f i e l d grows a charge density w i l l be induced on the surface of the bulb as determined by equation 2.2-12: This charge density w i l l grow u n t i l the internal f i e l d reaches the threshold f i e l d E Q . Once E Q i s reached an. avalanche breakdown w i l l occur and the e l e c t r i c f i e l d in the bulb w i l l be reset to 0. The time at which the avalanche occurs w i l l be designated t 0 . Note, the bulb's internal f i e l d i s the applied f i e l d plus the f i e l d produced by the induced surface charge. It w i l l be assumed the induced surface charge produces a uniform f i e l d which opposes the applied f i e l d . This problem w i l l constructed so i t reduces to the "switch on" problem discussed e a r l i e r , when "a" tends to ( a t ) z 2.12-1 *E- atd.(r) 2.12-2 2.2-12 - 66 -zero and t tends to i n f i n i t y . Consider t large so a l l transient solutions of equation 2.2-12 have disappeared. Furthermore, consider "a" small and t large such that the f i e l d in the bulb vanishes. The applied f i e l d changes so slowly that the f i e l d i s completely screened. Then the induced surface charge on the bulb i s : <5(r,t) - 6 ( r ) a t 2.12-3 where 6(r) i s the solution to the appropriate "switch on" problem. However, I am not interested in "a" small and-t large. Hence I w i l l assume that the induced surface charge, 6 ( r , t ) , has a transient which i s of the same form as the transients which occur in the -"switch on" problem, thus: 6(r,t) - «(r)(at-S(l-exp(-At))) 2.12-4 where 6 i s an undetermined c o e f f i c i e n t . B w i l l be determined by forcing 6(r) to s a t i s f y the "switch on" problem. 6(r) s a t i s f i e s the "switch on" problem i f equations 2.2-16 and 2.2-17 are v a l i d . Substitute equations 2.12-4 and 2.12-2 into equation 2.2-12 and equate growth and decay rates. The resulting equations are: 2.12-5 - 67 -g " ^ ; : : } B A + -o •o K 2.12-6 ^JAwcoR 2. 12-7 These equations are equivalent to equations 2.2-16 and 2.2-17 i f B=a/\. Note that the summation in equation 2.2-16 has been dropped since in a l l cases of interest there i s only one eigenvalue. Hence, the surface charge which is induced on the s h e l l due to an i n f i n i t e ramp i s : « ( r , t ) - 6( r ) (at - f-( 1-exp (-X t) ) ) 2.12-8 The e l e c t r i c f i e l d created by the f i r s t term in equation 2.12-8 exactly cancels the applied f i e l d . Thus the internal f i e l d , E, , i s equivalent to a surface charge 6 L ( r , t ) where: fiL(r,t) - - 6 ( r ) a - ( l - e x p ( - X t ) ) 2.12-9 Thus the internal f i e l d , E, , i s : E, - a - ( l - e x p ( - X t ) ) i 2.12-10 When the internal f i e l d reaches the threshold value, E D , an avalanche breakdown occurs and the internal f i e l d i s reset to 0. This occurs at time t 0, thus: E 0 - J(l-exp(-Xtj )>) 2.12-11 Rearrange 2.12-11 to obtain: - 68 -2.12-12 If one assumes the that the charge transport mechanisms are unaffected by the presence of the wall charges, then each subsequent breakdown w i l l be l i k e the f i r s t one. Hence the frequency of breakdown i s given by f B = i / t 0 . Thus: 2.12-13 This solution can be applied to a triangular wave. Consider the wave shown in figure 19. Let E A be the amplitude of the triangular wave, T the period of the wave and £ A the frequency of the wave. Then "a" i s : a - 2 f A E A 2.12-14 Substitute equation 2.12-14 into 2.12-13 to obtain: 2. 12-15 In equation 2.12-15 the corners of the triangular wave have been assumed not to a f f e c t , substantially, the pulse rate of the bulb. This assumption i s v a l i d at pulse rates much higher then the threshold pulse rate. Moreover, I am interested in high applied e l e c t r i c f i e l d s such that: - 69 -A E 0 Equation 2.12-15 can be series expanded for t h i s extreme. The results are: f B * ^ | ^ A - | when -Ala- << i 2.12-16 1 1 o * 2f A EA I am interested in sinusoidal applied f i e l d s . Hence, parameter "a" in equation 2.12-13 changes with time. Since "a" i s an important parameter in the derivation, I would l i k e to approximate the sine wave with a wave which has a similar slope and has the same amplitude. The simplest and most accurate approximation i s shown in figure 20. This approximate form has the same slope as the sine wave at the zero crossings but i s clipped off so the approximate wave has the same amplitude as the sine wave. It is cle a r , from figure 20, that: a - wfA E A 2.12-17 Furthermore, multiply the pulse rate f B by 2/ir to account for c l i p p i n g the peaks of the triangular wave. Hence equation 2.12-13 becomes: B it V **AE A/ 2.12-18 I am interested in high e l e c t r i c f i e l d s , thus equation 2.12-18 can be series expanded to obtain: - 70 -DEFINITION OF " a " FOR A TRIANGULAR WAVE Q _l UJ U -C_> DC H-O UJ _l 111 A s 2EA_ A E A / APPLIED FIELD , T f A = 7 \ a s 2 E A f / \ / I -A f \l 1 I. TIME FIGURE 19 DEFINITION OF " a " FOR A CLIPPED TRIANGULAR WAVE BEST APPROXIMATION TO A SINE WAVE E A 2 FIGURE 20 - 71 -f B * 2 f A | A - i when -i£a « x 2.12-19 The "reset screening" model, as seen from equation 2.12-19, predicts the offset of the c a l i b r a t i o n curves from the "zero conductivity curve" at high e l e c t r i c f i e l d s . Figure 21 shows the the o r e t i c a l "reset screening curve" and " s t a t i c screening curve" on the same plot as well as a t y p i c a l c a l i b r a t i o n curve of a bulb. For many bulbs the c a l i b r a t i o n curve w i l l follow the " s t a t i c screening curve" at low count rates as shown in figure 21. A careful examination of the derivation of the "reset screening" model w i l l explain t h i s phenomenon. The heart of the "reset screening" model i s that the transient solutions of equation 2.2-12 play an important r o l e . Let " t 0 " be the time between pulses. I f : A t o > > 1 2.12-20 then the transient solutions play no role in the operation of the bulb. Hence equation 2.12-4 i s not v a l i d which means the "reset screening" model i s not v a l i d . At threshold t 0«l/ 2 f A . Thus i f : 7 f A > 1 2.12-21 the c a l i b r a t i o n curve i s expected to follow the " s t a t i c screening curve" at low count rates. I f : - 72 -2.12-22 only the threshold point is expected to be on the " s t a t i c screening curve". Of course there i s a t r a n s i t i o n zone in which neither model applies. 2.13 LINKING OF THE "RESET SCREENING" AND "STATIC SCREENING" MODELS I now have two ways of c a l c u l a t i n g X from one c a l i b r a t i o n curve. F i r s t l y , I can fi n d X from the Y-intercept of the "reset screening curve": x * * Y S 2.13-1 where Y g i s the Y-intercept of the "reset screening curve" as shown in figure 21. Secondly, I can find X by talcing the ra t i o of the slopes of the "reset screening curve" to the " s t a t i c screening curve". From equation 2.11-12: where Ms i s the slope of the " s t a t i c screening curve". From equation 2.12-19: 2.13-2 2.13-3 where MR i s the slope of the "reset screening curve", hence: - 73 -RESULTS OF THE STATIC AND RESET SCREENING MODELS LU < LU CO I CL A C T U A L C A L I B R A T I O N CURVE V R E S E T S C R E E N I N G CURVE E A A 2 f A £ A - ± Eo ir S T A T I C S C R E E N I N G CURVE 2E A f A T I T / / A P P L I E D F I E L D / E, / / / / / Y 5 = A/ir FIGURE 21 - 74 -F i n a l l y , as shown e a r l i e r , A s h i f t , 0 , of the pulses describes t h i s r e l a t i o n s h i p : X • wTanG 2.13-4 can be determined by the phase at threshold. Equation 2.11-10 2.11-10 - 75 -CHAPTER 3 EXPERIMENTS 3.1 INTRODUCTION The purpose of the experiments performed in t h i s research was to evaluate the theoret i c a l models presented in the previous chapter. F i r s t l y I w i l l b r i e f l y describe the apparatus common to a l l the experiments. The apparatus i s described in more d e t a i l than given below in a paper by D.E. Friedmann et. a l . The apparatus used in t h i s experiment e s s e n t i a l l y consisted of two parts; a device to create uniform alternating e l e c t r i c f i e l d s , and a device to measure the magnitude of these f i e l d s . The f i e l d was created by applying transformer outputs to two sets of p a r a l l e l plate capacitors. For one set, the plate dimensions were 3 f t . by 3 f t . and plate separation ranged from 6 i n . to 1 f t . These plates were driven by a 7 kV transformer. The f i e l d was sinusoidal and the frequency range was 40 Hz to 1 kHz. The other set of plates were 8 f t . by 8 f t . and the plate separation ranged from 6 i n . to 3 f t . These plates were fed at 60 Hz by a 30 kV transformer. In both cases the transformer output was varied by using a variac for the input supply. The plate voltage was measured with the use of a 1000:1 voltage divider and a d i g i t a l v o l t meter. See Friedmann,D.E.-, Curzon,F.L., Feeley,M., Young,J.F. and Auchinleck,G. Rev. S c i . Instrum., Vol. 53, 1273 (1982) DEVICE FOR PRODUCING ELECTRIC FIELDS 120 V.A.C. 60 Hz FREQUENCY GENERATOR A M P L I F I E R P A R A L L E L P L A T E CAPACITOR TRANSFORMER T VOLTAGE D IV IDER FIGURE 22 DEVICE FOR MEASURING ELECTRIC FIELDS O P T I C A L F I B E R B U L B " Q -I I l P H O T O M U L T I P L I E R o O S C I L L O S C O P E FREQUENCY COUNTER FIGURE 23 - 78 -figure 22 for a diagram of the apparatus. The applied f i e l d , EA , can be calculated from the plate voltage and the plate separation. Calibrated bulbs were used to measure the amplitude of the e l e c t r i c f i e l d s . A ca l i b r a t e d bulb is one for which the plot of the pulse rate versus the magnitude of the applied e l e c t r i c f i e l d has been determined. The pulses of l i g h t were transmitted from the region near the bulb with a nonconducting o p t i c a l f i b e r . This ensured the f i e l d near the bulb was not substantially affected by the device used to measure the pulse rate of the bulb. The o p t i c a l fiber was connected to a photomultiplier whose amplified output was monitored by a frequency counter so as to determine the pulse rate of the bulb, f B. The photomultiplier output was also monitored with an os c i l l o s c o p e . See figure 23 for a diagram of the apparatus. 3.2 VERIFICATION OF THE "STATIC SCREENING" MODEL The " s t a t i c screening" model can be tested by measuring the screening of alternating e l e c t r i c f i e l d s inside conducting tubes. For example, consider a long conducting tube with no ends on i t . Furthermore, define a coordinate system such that the axis of the tube i s p a r a l l e l to the Z-axis as shown in figure 24. Let the time constant for the response of the tube be Aj. when the applied f i e l d i s perpendicular to the axis of - 79 -TUBE ORIENTATION WITH RESPECT TO CARTESIAN COORDINATES X TUBE L Y FIGURE 24 the tube and A „ when the f i e l d i s p a r a l l e l to the axis. These time constants have been derived in the previous chapter. \ ± can be assumed to be equal to the time constant of the response of an i n f i n i t e cylinder with the applied f i e l d perpendicular to i t s axis. Hence, from equation 2.10-7, X X i s : This re s u l t i s v a l i d for an i n f i n i t e cylinder, hence one would expect the response of a f i n i t e cylinder to be modified near the ends of the cylinder. However, i f the cylinder i s long enough the response near the centre of the f i n i t e cylinder w i l l be similar to that of an i n f i n i t e c y l i n d e r . It was found that spheroids could be considered i n f i n i t e l y long when L/D>3; i t w i l l be assumed th i s also Q 3.2-1 - 80 -applies to tubes. Hence equation 3.2-1 i s v a l i d for conducting tubes with L/D>3. The di s t o r t i o n s of the e l e c t r i c f i e l d , near the ends of the tube, w i l l extend a distance approximately equal to i t s diameter into the tube. This i s consistent with the assumption that tubes with L/D>3 respond l i k e i n f i n i t e c ylinders. X„ can be assumed to be equal to the time constant of the response of a long spheroid with the applied f i e l d p a r a l l e l to i t s axis. As in the previous case, the lengths of the spheroid and cylinder are equal and the diameter of the tube i s equal to the maximum diameter of the spheroid. Hence, from equation 2.6-4, X„ i s : There i s some error in t h i s assumption due to the dra s t i c differences in the geometries of the spheroid and tube at thei r ends. However, i t i s clear that as L/D of these objects becomes larger the differences in their time constants is reduced. It i s expected that the e l e c t r i c f i e l d , near the ends of the tube, w i l l be distorted approximately one diameter in length in from the ends of the tube. Near the centre of the tube the time constant of the response of the e l e c t r i c f i e l d w i l l be approximately the same as the time constant in the centre of a long spheroid. A spheroid can be considered long when L/D>3. The time constant for a tube w i l l be assumed to the same as a long spheroid's when L/D>3. 3.2-2 - 81 -The r a t i o of the time constants for the f i e l d applied p a r a l l e l to the axis of the tube to the time constant for the f i e l d applied perpendicular to the axis i s : This quantity i s convenient since i t eliminates the conductivity from the analysis. Furthermore i t can be measured experimentally. A „ L 2 / X 1 D 2 i s plotted against L / D in figure 25. Equation 3.2-3 i s a the o r e t i c a l prediction of X|, / Xx. The r a t i o of the time constants can also be determined experimentally by measuring the screening of alternating e l e c t r i c f i e l d s inside conducting tubes. Consider equation 2.11-9: Here E, and E A are the magnitudes of the internal and external f i e l d s respectively and u i s the frequency of the f i e l d . This equation can be rearranged to obtain: 3.2-3 2.11-9 3.2-4 Thus, with the use of equation 3.2-4, X„/xA i s : 3.2-5 A PLOT OR THE THEORETICALLY PREDICTED X,, /Xx - 83 -Here (E A/E, ) i s the r a t i o of the magnitudes of the applied f i e l d to the internal f i e l d when the f i e l d i s applied p a r a l l e l to the axis of the cylinder and (E A/E, ) x i s the same r a t i o for the case when the f i e l d i s applied perpendicular to the axis. These quantities can be measured experimentally. Hence XM/Xj. can be determined experimentally with equation 3.2-5 and t h e o r e t i c a l l y with equation 3.2-3. F i r s t l y I w i l l describe how (E A/E|) ± was measured. The tube was held between the plates of the p a r a l l e l plate capacitor with plexiglass holders as shown in figure 26. The plexiglass holders did not a f f e c t the experimental results since the materials which made up the tubes were much more conducting than the p l e x i g l a s s . The plate separation was at least three times the diameter of the tube, and the tube was centred between the plates. This was done'to ensure the applied e l e c t r i c f i e l d , E A , was uniform. Furthermore, the distance S defined in figure 26 was greater than 3D to ensure the experimental results were not influenced by the ends of the p a r a l l e l plate capacitor. The e l e c t r i c f i e l d monitor (a bulb) was held in the centre of the tube with with a plexiglass holder as shown in figure 26. The equipment was b u i l t such that the tube could be moved in and out of the plates without disturbing the bulb. This i s important because the c a l i b r a t i o n curve of the bulb can be affected i f the bulb i s disturbed. F i r s t l y the tube was removed from the plates and the bulb c a l i b r a t e d . Next the tube was placed into position without disturbing the bulb. Some tubes had seams which ran - 84 -p a r a l l e l to their axes; these were placed with the seam oriented as shown in figure 26. The applied f i e l d was increased u n t i l the bulb was operating. From the c a l i b r a t i o n curve the internal f i e l d , E, , was determined; and the applied f i e l d , E A , was determined from the plate voltage and plate separation. Hence (E A/E, ) x was measured. To measure (E A / E , ) M the apparatus was rearranged to the configuration shown in figure 27. The separation of the plates was such that the ends of the tube were at least one tube diameter away from the plates. This was done to ensure the applied f i e l d was r e l a t i v e l y uniform. Furthermore, the distance S, defined in figure 27, was greater than 2D+L to ensure the experimental results were not influenced by the edges of the plates. The tube could be removed from the plates without disturbing the bulb. The procedure for measuring (E A/E, ),, was i d e n t i c a l to the procedure for measuring (E A/E, )x as described e a r l i e r . Tubes were made either from paxolin or neoprene since the c o n d u c t i v i t i e s of these materials are in the range which allows E A/E, to be measured with the apparatus. The conductivity of the material must be such that E, /E A i s somewhere between 0.05 and 0.95 because the e l e c t r i c f i e l d monitor i s , at best, accurate to within 5%. A diameter of approximately 3 inches was chosen for the tubes, since t h i s dimension i s large in comparison to the bulb diameter and small compared to the plate separation. The neoprene tubes were constructed from f l a t sheets of neoprene which ranged in thickness from 1/8 to 1/4 of an CONFIGURATION OF THE APPARATUS USED TO MEASURE (EA/E, ) ± L/2 O P T I C A L F I B E R X SEAM B U L B zzzz V * D/2 T v t >>>>>>>> _\ > > \ RUBBER TUBE D i . 3D S > 3D 7" P L E X I G L A S S HOLDERS SEAM T U B E FIGURE 26 CONFIGURATION OF THE APPARATUS USED TO MEASURE (E A/H, )„ T D D D -*» H 1 RUBBER T U B E B U L B 7 * O P T I C A L F I B E R 7 C A P A C I T O R P L A T E "7 P L E X I G L A S S HOLDERS X S > L+2D FIGURE 27 - 87 -inch. The tubes were wrapped with black e l e c t r i c a l tape in order to maintain the c y l i n d r i c a l shape. The e l e c t r i c a l tape turned out to be much less conducting than the neoprene, hence i t did npt affe c t the experimental r e s u l t s . Each neoprene tube had a seam which ran p a r a l l e l to the axis of the tube. The seam was oriented in the manner shown in figure 26, since by symmetry there was no voltage drop across i t . In t h i s way the joint did not a f f e c t the current flow. To check the theory as thoroughly as possible (E A /E, )„ and (E A /E , ) 1 were measured over a range of frequencies of the applied f i e l d (40 Hz - 1 kHz), and for 3 < L/D < 11. Theory requires L/D>3. The upper l i m i t on the length to diameter r a t i o was set by the need to measure E, to an accuracy of better than 5%. The res u l t s of the measurements are: - 8 8 -TABLE I, STATIC SCREENING DATA L/D MATERIAL ( E A / E , )„ ( EA / E , ) L Ui D^/EX Ui D * / T H 3 paxolin 4 0 Hz 1 . 2 1 2 1 . 7 9 2 4 . 1 5 3 . 5 0 3 paxolin 6 0 Hz 1 . 0 9 3 1 . 6 1 0 3 . 1 4 3 . 5 0 3 paxolin 6 0 Hz 1 . 1 0 6 1 . 5 8 0 3 . 4 8 3 . 5 0 3 paxolin 1 2 0 Hz 1 . 1 1 1 1 . 4 5 6 4 . 1 2 3 . 5 0 3 paxolin 1 6 0 Hz 1 . 1 0 4 1 . 5 8 5 3 . 4 2 3 . 5 0 3 paxolin 2 0 0 Hz 1 . 0 9 6 1 . 5 9 0 3 . 2 7 3 . 5 0 3 neoprene 4 0 0 Hz 2 . 8 9 0 7 . 2 0 0 3 . 4 2 3 . 5 0 3 neoprene 7 0 0 Hz 2 . 4 4 0 5 . 8 0 0 3 . 5 1 3 . 5 0 3 neoprene 1 0 0 0 Hz 2 . 2 7 0 5 . 0 0 0 3 . 7 4 3 . 5 0 5 neoprene 6 0 Hz 1 . 5 6 3 5 . 1 8 2 5 . 9 7 5 . 8 1 5 neoprene 6 0 Hz 1 . 7 4 2 5 . 4 6 4 6 . 6 4 5 . 8 1 1 1 . 3 neoprene 6 0 Hz 2 . 8 0 0 3 2 . 4 3 1 0 . 3 9 . 5 1 1 1 . 3 neoprene 6 0 Hz 1 . 8 5 0 1 7 . 0 8 1 1 . 7 9 . 5 1 (XM L z/Xi D Z ) E X was calculated using equation 3 . 2 - 5 and (X„ L2/Xi D 2 ) T H was calculated using equation 3 . 2 - 3 . Both of these quantities are plotted against L/D in figure 2 8 . The agreement between the theory and the experimental points i s reasonable. Notice that X,,/Xj. i s r e l a t i v e l y frequency independent as expected. It turned out the experimental points were r e l a t i v e l y d i f f i c u l t to obtain. F i r s t l y , the tube material had such a small conductivity that merely handling the tubes affected the conductivity enough to a l t e r the r e s u l t s . Hence much - 90 -care was taken to ensure the tubes were very clean. Moreover, the humidity affected the r e s u l t s . For example, for the tube with L/D = 11.3, two runs where taken at 60 Hz. The quantity (E A /E, )A and ( E A / E , ) „ changed dramatically between these runs because the rubber was washed between runs; t h i s changed the conductivity of the rubber. The runs at 60 Hz with L/D = 3 and L/D = 5 were also influenced by changes in humidity. Fortunately the quantity Xi,/X± i s independent of the conductivity of the rubber, hence the experimental results are r e l a t i v e l y unaffected. However, changes in the conductivity due to cleanliness and humidity can result in some non-isotropic conduction. This e f f e c t i s l i k e l y not important since the re s u l t s agree reasonably well with the theory, but i t i s n o n - s t a t i s t i c a l in nature hence i t i s d i f f i c u l t to predict the average error due to t h i s e f f e c t . Moreover, i t was d i f f i c u l t to maintain the neoprene tubes in a c y l i n d r i c a l shape since the tubes were not r i g i d . For tubes even as l i t t l e as 10% out of round the experimental results were affected by as much as 30%. Unfortunately a r i g i d insulating support structure could not be used to maintain the shape of the neoprene, because i t was found s t a t i c charges tend to b u i l d up on the contact areas between the insulating structure and the neoprene. This affected the experimental r e s u l t s . The paxolin tubes were s o l i d , therefore they were not affected by t h i s problem. The e l e c t r i c f i e l d monitor was another source of error, since the bulbs are s l i g h t l y unstable. The pulse rate of a bulb varied as much as 10% from run to run at a given - 91 -applied f i e l d strength and frequency. Thus, due to the nature of the experiment, the results should be taken as q u a l i t a t i v e . The assumptions required to derive the s t a t i c screening model, which i s the theory tested in t h i s experiment, are; the frequency of the applied f i e l d i s low so magnetic e f f e c t s can be neglected, current can only flow on the s h e l l surface and the s h e l l possesses is o t r o p i c and Ohmic conduction. The f i r s t two assumptions were v a l i d in this experiment. The remaining two assumptions depend on the material used to construct the tubes. However, the materials used do appear to have isotropic and Ohmic conduction because the data agreed reasonably well with the theory. Furthermore, i t seems that the assumption that a long, open ended, tube can be treated as a long spheroid, i s a v a l i d assumption. It i s interesting to plot the conductivity of the tube material as a function of frequency, since the conductivity seems to be dependent on frequency. From equation 3.2-4: Furthermore, from equation 3.2-1, Ai i s : 3.2-1 3.2-6 The diameter of the L/D = 3 neoprene tube was 10 cm. and the - 92 -CONDUCTIVITY OF PAXOLIN PLOTTED AGAINST FREQUENCY 1 3 0 0 0 f © - EXPERIMENTAL POINT e J7 2000 I (0 I ? u a 1000 + 8 50 100 150 «/2tr (Hz) 200 FIGURE 29 - 93 -CONDUCTIVITY OF NEOPRENE PLOTTED AGAINST FREQUENCY 4 © - E X P E R I M E N T A L POINT 60, 000 + w a 40, 000 + 20,000 + 400 800 w/2« (Hz) FIGURE 30 - 94 -diameter of the paxolin tube was 6 cm. The data from Table 1 was used to obtain fl/e0 with equation 3.2-6. The results are: TABLE II, CONDUCTIVITY OF PAXOLIN AND NEOPRENE M.K.S. units MATERIAL fA n/e0 paxolin 40 Hz 23 paxolin 60 Hz 29 paxolin 60 Hz 28 paxolin 120 Hz 48 paxolin 160 Hz 74 paxolin 200 Hz 93 neoprene 400 Hz 1790 neoprene 700 Hz 2520 neoprene 1000 Hz 3080 This data i s plotted in figures 29 and 30. 3.3 EXPERIMENTAL VERIFICATION OF THE FREQUENCY DEPENDENCE OF THE CONDUCTIVITY OF NEOPRENE 0/e was measured as a function of frequency by experimentally determining the magnitude and phase of the e l e c t r i c current flowing in f l a t sheets of neoprene due to a voltage applied across the neoprene. For example, consider a - 95 f l a t rectangular piece of neoprene with an alternating e l e c t r i c potential applied across i t as shown in figure 31. NEOPRENE DIMENSIONS FLAT PIECE OF NEOPRENE i FIGURE 31 The f i e l d internal to the neoprene, E , , i s related to the applied f i e l d by equation 2.11-8: E \ V. 1 - 2.11-8 0k where z indicates the d i r e c t i o n of the applied f i e l d , VA i s the voltage drop across the neoprene and L i s the length of the neoprene sheet as shown in figure 31. The internal f i e l d , Is, , w i l l drive a surface current J on the neoprene. 3 i s related to E \ through Ohms law: - 96 -j -0E A 3.3-1 where Q i s the surface conductivity of one sheet of neoprene ( i . e . Ohms per square). It turns out the experimental results are consistent with the assumption that current only flows on the surface of the neoprene. Substitute equation 3.3-1 into equation 2.11-8 to obtain: * J V » -• " * = ( i - 4)L 3 - 3 ? 2 To increase the current flow through the neoprene many sheets of t h i s material were bundled together since the t o t a l current flow through the neoprene i s proportional to the number of sheets. The t o t a l current, I, flowing through "n" sheets of neoprene i s : I • Jn£ 3.3-3 where J i s the magnitude of the surface current , £ i s defined in figure 31 and "n" i s the number of neoprene sheets. Thus, from equation 3.3-2: I VA -2 3.3-4 Let 0 be the phase difference between the applied f i e l d and the current, then from equation 3.3-2: 6 - Tan 1-^ 3.3-5 Hence equation 3.3-4 becomes: - 97 -LI V A£nCos9 3.3-6 Cl e a r l y the time constant, X , of a f l a t sheet i s : The time constant i s dependent on the charge to charge interaction of the electrons as they flow in the rubber. Thus e was used rather then e0 in equation 3.3-7 since "n" was quite large, which means most of the current flow was within the d i e l e c t r i c . Equation 3.3-5 becomes: ^ = LuTane 3.3-8 Substitute equation 3.3-8 into equation 3.3-6 to obtain: I E " VA -twSine 3.3-9 Thus, e , ft and ft/e can be calculated with equations 3.3-9, 3.3-6 and 3.3-8 respectively i f the magnitude and phase of the voltage across the neoprene sheets i s known as well as the magnitude and the phase of the current through the sheets. The apparatus was arranged to the configuration shown in figure 32. The f l a t neoprene sheets were held firmly between the capacitor plates. A mylar sheet was wrapped around the neoprene to create an insulating b a r r i e r which allows only c a p a c i t i v e l y coupled current to flow in the c i r c u i t . Of course there was a contribution to the current in the c i r c u i t due to the capacitance of the plates. It - 98 -CONFIGURATION OF THE APPARATUS USED TO MEASURE fl/e b H.V. O O S C I L L O S C O P E FIGURE 32 - 99 -turns out t h i s current i s n e g l i g i b l e . The magnitude and phase of the voltage VH was measured with an oscilloscope as shown in figure 32. The plate voltage was then calculated with VH and the values of the r e s i s t o r s which form the voltage d i v i d e r . Thus the plate voltage was: VA= 1000V H 3.3-10 The magnitude and phase of voltage V, was also measured with the oscilloscope. Thus the current through the neoprene was: 1= V, /R1 3.3-11 It turns out V, was sinusoidal when VH was sinusoidal, hence the conductivity of the neoprene was Ohmic. The magnitudes of the resistances in the c i r c u i t were chosen such that the experimental results were not affected by t h e i r presence. The dimensions of the neoprene sheets, defined in figure 31, were: 1=0.151m L = 0.1 25m D = 0.0050m The r e s u l t s of a run with three sheets of neoprene placed between the plates are: -100 -TABLE I I I , CONDUCTIVITY OF NEOPRENE (RUN WITH 3 SHEETS) M.K.S. units n U / 2 T e / e 0 OxlO"8 O/e n/e 0 3 95 Hz 1 1 .9 1 .8 171 2030 3 142 Hz 10.6 2.1 224 2370 3 191 Hz 9.8 2.4 276 2700 3 238 Hz 8.6 2.6 341 2930 3 286 Hz 8.0 3.0 423 3380 3 333 Hz 7.8 3.3 477 3720 3 1000 Hz 5.9 7.2 1377 81 20 The results of a run with f i v e sheets of neoprene placed between the plates are: TABLE IV, CONDUCTIVITY OF NEOPRENE (RUN WITH 5 SHEETS) M.K.S. units n <i)/2* OxlO"8 fl/e n/e 0 5 95 Hz 11.5 2.0 196 2250 5 142 Hz 10.5 2.5 269 2820 5 191 Hz 9.6 2.8 329 3158 5 238 Hz 9.0 3.0 376 3380 5 286 Hz 8.6 3.2 420 3600 5 333 Hz 8.3 3.4 462 3830 5 1000 Hz 6.0 7.2 1354 8120 -101 -e/e 0 , a and ft/c are plotted against u>/2» in figures 33, 34 and 35 respectively. As expected, the quantities measured,c/e o r a and fl/e do not depend on the number of sheets used in the run. Any differences were l i k e l y due to the large errors involved in t h i s experiment. For example, the conductivity of the neoprene was very sensitive to the cleanliness of the neoprene and the humidity. Merely handling the neoprene from run to run affected the r e s u l t s . An e f f o r t was made to keep the neoprene as clean as possible, however th i s was s t i l l a major source of error. Furthermore t h i s error was n o n - s t a t i s t i c a l in nature ( i . e . the conductivity of the neoprene evolves with time) hence i t was inappropriate to estimate the magnitude of the error. Thus, these re s u l t s should be taken as q u a l i t a t i v e . These re s u l t s , tables III and IV, can be compared to the re s u l t s from the previous section for the L/D=3 neoprene tube, table I I . As can be seen from these tables, 0/e0 was measured to be the same order of magnitude in both experiments even though the tube was constructed from neoprene which was a di f f e r e n t thickness and was from a di f f e r e n t source. F i n a l l y , in order for equations 3.3-9, 3.3-6 and 3.3-8 to be v a l i d , the " s t a t i c screening" model must be v a l i d . The assumptions required to derive the " s t a t i c screening" model are; the frequency of the applied f i e l d must be low so magnetic e f f e c t s can be neglected, no gaseous breakdowns occur in or near the conducting material and the conducting material must possess is o t r o p i c and Ohmic conduction. The f i r s t two assumptions were v a l i d in t h i s experiment. -102 -EXPERIMENTAL e/e 0 , FOR NEOPRENE, PLOTTED AGAINST FREQUENCY © - RUN WITH 5 SHEETS OF NEOPRENE + - RUN WITH 3 SHEETS OF NEOPRENE 12 + + €> o u 8 + © + © + e + 4 + •+ 4-100 200 o)/2w (Hz) FIGURE 33 300 -103 -EXPERIMENTAL fl, FOR NEOPRENE, PLOTTED AGAINST FREQUENCY 3 + 6 XI O co i o 7 2 a 1 + © - RUN WITH 5 SHEETS OF NEOPRENE + - RUN WITH 3 SHEETS OF NEOPRENE © + e + + + + 100 200 w/2w (Hz) —t-* 300 FIGURE 34 - 1 0 4 -EXPERIMENTAL n/e, FOR NEOPRENE, PLOTTED AGAINST FREQUENCY + 400 3004-e v f U a 200 + l O O f © . RUN WITH 5 SHEETS OF NEOPRENE + - RUN WITH 3 SHEETS OF NEOPRENE © + + 100 200 300 W/2w (Hz) FIGURE 35 -105 -Furthermore the neoprene possessed Ohmic conduction since, V, was sinusoidal when V H was sinusoidal. It i s d i f f i c u l t to determine whether the material possessed isotropic conduction, however, t h i s i s l i k e l y a v a l i d assumption. The important conclusion i s that fl/e0 seems to have a frequency dependence, at low frequencies ( i . e . 0 •*• 1000 Hz), as measured in the previous section. 3.4 EXPERIMENTAL VERIFICATION OF THE THEORY WHICH EXPLAINS THE FORM OF THE CALIBRATION CURVES OF THE BULBS In chapter 2, section 13, the " s t a t i c screening" model was combined with the "reset screening" model to explain the form of a t y p i c a l c a l i b r a t i o n curve of a bulb, as shown in figure 21. The " s t a t i c screening" model i s v a l i d below threshold and sometimes s l i g h t l y above threshold. The assumptions required to derive the model are; the frequency of the applied f i e l d i s low so magnetic ef f e c t s can be neglected, no avalanche breakdowns occur and the bulb's surface possesses isotropic and Ohmic conduction. The results of the derivation are the pulse r a t e , f B , in a sinusoidal f i e l d i s : Furthermore, the phase s h i f t of the pulses at threshold i s : 2.11-12 6 • Tan 2.11-10 where e i s defined in figure 18. -106 -The "reset screening" model i s v a l i d at high pulse rates. The assumptions required to derive t h i s model are the same as the ones required to derive the " s t a t i c screening" model, except, i t i s assumed many avalanche breakdowns occur during one cycle of the applied f i e l d . Hence, t h i s model i s v a l i d at high f i e l d s . According to t h i s model the pulse r a t e , f B , i s : fB^ " * t * - * 2.12-19 where the applied f i e l d used to derive t h i s equation is an approximation to a sinusoidal f i e l d as shown in figure 20. These models were combined in chapter 2, section 13, to obtain three ways to measure X. They are: 1 2.13-4 X n Y s 2.13-1 X - wtanG 2.11-10 where Ms i s the slope of the " s t a t i c screening curve", M„ i s the slope of the "reset screening curve" and Ys i s the Y-intercept of the "reset screening curve" as shown in figure 21. 9 i s defined in figure 18.X depends d i r e c t l y on the conductivity of the bulb's surface. The conductivity of many types of glass i s too low to observe the effect described above. Hence, soda-lime glass was used to construct the bulbs for t h i s experiment since i t i s a r e l a t i v e l y conductive glass. One of these bulbs was placed -107 -between the capacitor plates and c a l i b r a t e d . This bulb was 4 cm. in diameter and contained argon at 1.6 t o r r . The experimentally obtained c a l i b r a t i o n curves, f B plotted against E A , are presented in graphical form in figures 36, 37, 38 and 39 for various frequencies of the applied f i e l d . MS, MR and Y S as measured from th i s data are presented in table V. TABLE V, BULB DATA w/2* M R ( T V ) " • ( i v ) Y S (Hz) 40 Hz 11.7 3.5 280 60 Hz 21.7 6.5 460 80 Hz 25.5 9.7 440 100 Hz 31.4 12.7 520 X may be calculated in two ways with t h i s data, with equation 2.13-1 and with 2.13-4. Let X M be the experimentally obtained X as calculated with equation 2.13-4 and l e t XY be the experimentally obtained X as calculated with equation 2.13-1. The results are presented in table VI. -108 -SODA-LIME BULB CALIBRATION CURVE FOR 40Hz FIGURE 36 -109 -SODA-LIME BULB CALIBRATION CURVE FOR 60Hz k>/2w » 60Hz FIGURE 37 — 110 — SODA-LIME BULB CALIBRATION CURVE FOR 80Hz u/2« = 80Hz FIGURE 38 -111 -SODA-LIME BULB CALIBRATION CURVE FOR 100Hz FIGURE 39 -112-TABLE VI, EXPERIMENTALLY OBTAINED TIME CONSTANTS u/2* A y (s"1) A y / A M 40 Hz 800 880 1.10 60 Hz 1200 1450 1.21 80 Hz 1220 1 380 1.13 100 Hz 1420 1630 1.15 The magnitude of the error in the experimentally obtained X i s d i f f i c u l t to determine since the pulse rate, f B , at a given frequency, u , and applied f i e l d , E A , would vary approximately 10% from run to run. Hence, the c a l i b r a t i o n curves are not e n t i r e l y reproducible. Furthermore, t h i s change from run to run in X i s not necessarily a s t a t i s t i c a l f l u ctuation, these bulbs are s l i g h t l y unstable hence their properties evolve with time. Thus, the experimental errors were not necessarily reduced by averaging the results over many runs. From table VI, one can see the agreement between Xy and X M i s quite reasonable. Xv i s , on average, approximately 15% higher than X M. There are two possible explanations for t h i s error. The glass which makes up the bulbs i s s l i g h t l y d i s torted in the region where the bulb was f i l l e d with gas and sealed o f f . This means the conductivity of the surface is not is o t r o p i c in that region. Hence some error in X i s expected. Fortunately t h i s area i s a small f r a c t i o n of the -113-t o t a l surface area. Furthermore, equation 2.12-19 was derived with an applied f i e l d which only approximated a sinusoidal f i e l d . This applied f i e l d i s shown in figure 20. This i s also a source of error. A l l other assumptions used to derive the "reset" and " s t a t i c screening" models were considered v a l i d . As a f i n a l t e s t , the phase of the pulses at threshold was measured and compared to the th e o r e t i c a l prediction: 0 = Tan - 2.11-10 The plate voltage and the bulb's pulses were observed at the same time with a storage oscilloscope as shown in figure 22 and 23. The applied e l e c t r i c f i e l d was in phase with the plate voltage. It was found that the phase of the pulses fluctuated. Hence, the oscilloscope was set to the envelope 16 mode, where the envelope 16 mode overlays 16 signals on the screen and stores them. Thus, one could see the pulses bunching up into groups as shown in figure 40. These groups of pulses are wide in comparison to the width of a period. Hence, i t i s d i f f i c u l t to obtain an average phase angle from these traces. Instead I w i l l substitute the experimentally obtained X M and A Y into equation 2.11-10 and obtain the values of e which are consistent with the c a l i b r a t i o n curve data. The position of the calculated 9 w i l l then be indicated on the pictures of the experimentally obtained 9 . Let 9 M be the results calculated from X M and 9 Y be the results calculated from A Y . These results are presented in table VII. -114-TABLE VII EXPERIMENTALLY OBTAINED PHASE ANGLES «/2ir eM eY QAVG 40 Hz 73° 74° 73.5° 60 Hz 73° 75° 74.0° 80 Hz 68° 70° 69.0° 100 Hz 66° 69° 67.5° The position of 6AVG i s indicated on the pictures of the experimentally obtained 6 in figure 40. As expected, the groups of pulses are more or less centered around QAVG . The " s t a t i c screening" model was derived assuming no avalanche breakdowns occur. The f i e l d was above threshold when 9 was measured. Thus, the " s t a t i c screening" model sta r t s to breakdown. This could be the cause of the fluctuations in 6. -1 15 PHASE SHIFT OF THE SODA-LIME BULB'S PULSES AT THRESHOLD THE TRACES WERE OBTAINED BY SUPERIMPOSING 16 SWEEPS AT THE THRESHOLD F I E L D THE BLACK VERTICAL LINES ARE eAVG AS PREDICTED BY A M AND AY PULSES A P P L I E D F I E L D e AVG u/2v « 40Hz \ ... i i i- — i — — — t u/2* « 60Hz u/2t * 80Hz M / 2 I T « lQQHz TIME FIGURE 40 -116-CHAPTER 4 CONCLUSIONS 4.1 INTRODUCTION The results presented in thi s thesis concern research directed towards understanding the operation of the g a s - f i l l e d glass bulbs, which emit pulses of l i g h t when placed in an alternating e l e c t r i c f i e l d of s u f f i c i e n t magnitude. In p a r t i c u l a r , the eff e c t of the conductivity of the bulb's s h e l l was studied. Two t h e o r e t i c a l models were developed to explain the influence of conducting s h e l l s on the c a l i b r a t i o n curves for electrodeless breakdown at low frequencies ( < 1kHz ). These res u l t s are of p a r t i c u l a r significance since such bulbs are used in devices which monitor environmental e l e c t r i c f i e l d s . It was found that the conductivity of the glass, from which the bulbs are made, can be affected by the environmental conditions. In p a r t i c u l a r , the humidity and the cleanliness of the glass a f f e c t the conductivity. The t h e o r e t i c a l models predict this w i l l lower the accuracy of the e l e c t r i c f i e l d monitors. Thus, i t is important to design monitors in which the bulbs are shielded from the environment such that the changes in their conductivity are minimized. Experimental v e r i f i c a t i o n of the th e o r e t i c a l models was observed. Furthermore a method of measuring the conductivity of some materials at low frequencies ( < 1 kHz ) was developed. The th e o r e t i c a l models and the experiments are summarized in more d e t a i l below. - 1 1 7 -4.2 CONCLUSIONS It turns out that i f the conductivity of the glass, which forms the bulb, i s ignored then the the o r e t i c a l c a l i b r a t i o n curve of the bulb i s : F B - E 0 7 2.1-4 as shown in chapter 2, section 1. This curve i s plotted in figure 41 and i s c a l l e d the "zero conductivity curve". Furthermore t h i s theory predicts that the phase s h i f t of the pulses at threshold should be as indicated in figure 42. However i t was found that many c a l i b r a t i o n curves are l i k e the one shown in figure 41, which i s l a b e l l e d the "observed c a l i b r a t i o n curve". Furthermore the pulses were phase shi f t e d at threshold. These findings were explained by allowing the bulb's s h e l l to be s l i g h t l y conducting, where s l i g h t l y conducting means the conductivity of the s h e l l w i l l noticeably a f f e c t the in t e r n a l f i e l d of the bulb with respect to the external f i e l d , but not screen i t out completely. This i s due to current flow in the glass which modifies charges on the bulb surface. Two models are required to explain the observed re s u l t s , each i s v a l i d in a di f f e r e n t regime of the bulb's operation. Both models contain one parameter, the conductivity of the surface. The f i r s t model, the " s t a t i c screening" model, i s v a l i d below threshold and possibly s l i g h t l y above threshold. However i t c e r t a i n l y i s not v a l i d when f B i s much greater than i t s threshold value. The assumptions required to derive RESULTS OF THE PROPOSED THEORETICAL MODELS OBSERVED C A L I B R A T I O N A P P L I E D F I E L D / E A * FIGURE 41 PHASE OF THE PULSES AT THRESHOLD FOR A NON-CONDUCTING BULB FIGURE 42 -119-t h i s model are; the frequency of the applied f i e l d i s low so that magnetic effects can be neglected, no avalanche breakdowns occur in the bulb and the bulb's surface possesses isotropic and Ohmic conduction. It turns out t h i s model predicts E, is reduced and phase shifted with respect to EA . Hence this model predicts that the slope of the c a l i b r a t i o n curve is reduced and the pulses are phase shi f t e d at threshold. These e f f e c t s are a function of the conductivity of the s h e l l . This curve i s c a l l e d the " s t a t i c screening curve", and i s shown in figure 41. The second model, the "reset screening" model, predicts the form of the c a l i b r a t i o n curve at high pulse rates. The assumptions required to derive t h i s model are the same as the assumptions required to derive the " s t a t i c screening" model except i t is assumed that many avalanche breakdowns occur during one cycle of the applied f i e l d . The t h e o r e t i c a l curve predicted by t h i s model i s shown in figure 41 and i s c a l l e d the "reset screening curve." At high count rates, the t h e o r e t i c a l curve i s p a r a l l e l to the "zero conductivity curve" but offset from i t as expected. Moreover, the offset of the "reset screening curve" from the "zero conductivity curve" only depends on the conductivity of the glass. The conductivity of the glass can be calculated in two independent ways with these models. F i r s t l y , the conductivity can be calculated from the r a t i o of the slopes of the "reset screening curve" to the " s t a t i c screening curve". Secondly, the conductivity can be calculated from the offset of the "reset screening curve" from the "zero -120 -conductivity curve." Thus, t h i s single parameter can be measured in two ways. Hence, the self-consistency of the models could be demonstrated experimentally. In chapter 2, the " s t a t i c screening" and "reset screening" models were derived. These models were applied to several geometries. They include; spherical s h e l l s , i n f i n i t e c y l i n d r i c a l s h e l l s and long spheroidal s h e l l s . In chapter 3, section 2, the " s t a t i c screening" model was tested d i r e c t l y . This was accomplished by measuring the screening of alternating e l e c t r i c f i e l d s inside conducting tubes. Two measurements were required; the screening was measured when the f i e l d was applied p a r a l l e l to the axis of the tube and when the f i e l d was applied perpendicular to the axis. With these measurements, the r a t i o of the time constants of the responses was calculated. This result was compared to the t h e o r e t i c a l r a t i o predicted by the " s t a t i c screening" model. The agreement between the theory and the experimental results was good as shown in figure 28. In t h i s experiment i t was found that the conductivity of neoprene was frequency dependent as shown in figure 30. The frequency dependence of the conductivity of neoprene was measured using another technique in chapter 3, section 3. This was accomplished by measuring the magnitude and the phase of an alternating voltage applied across f l a t sheets of neoprene as well as the resulting current through the sheets. With these quantities, the conductivity of the neoprene was calculated with the use of the " s t a t i c screening" model. The results were consistent with the -121 -res u l t s from the previous section. F i n a l l y , the " s t a t i c screening" model and the "reset screening" model were tested simultaneously as described in chapter 3, section 4. The c a l i b r a t i o n curve of a gas f i l l e d bulb was found to resemble the one shown in figure 41. The t h e o r e t i c a l " s t a t i c screening curve" and the theore t i c a l "reset screening curve" were f i t to the c a l i b r a t i o n curve. The parameters obtained from t h i s f i t were used to calculate the conductivity of the glass in two independent ways. It turned out these values were consistent. Furthermore the phase of the pulses at threshold were consistent with the " s t a t i c screening" model. Therefore the " s t a t i c " and "reset screening" models are consistent, hence they explain the form of the c a l i b r a t i o n curve of t h i s bulb. Thus the models developed in this thesis to take account of the e f f e c t of leakage current on the bulb's surface describe, very well, the form of the c a l i b r a t i o n curves obtained for electrodeless breakdown at low frequencies ( < 1 kHz). These results are important since such bulbs are used in e l e c t r i c f i e l d monitors. Because the conductivity of these bulbs i s dependent on environmental conditions, portable testing devices should be developed which, measure the bulb's threshold, obtain the bulb's c a l i b r a t i o n curve from the value of the threshold f i e l d with the t h e o r e t i c a l models and feeds t h i s information back into the f i e l d monitor. Periodic use of the testing device w i l l increase the accuracy of the monitor by correcting for the changing conductivity of the bulb. -122 -BIBLIOGRAPHY 1Harries fW.L. and Von Engel,A., Proc. Phys. S o c , London, Sect. B64,915(1951) 2 Meek,J.M., and Craggs,J.D., E l e c t r i c a l Breakdown of Gases , John Wiley and Sons, New York(1978) 3 Morse,P.M. and Feshback,H., Methods of Theoretical Physics McGraw-Hill Book Company, New York ( 1 953") 4 Jackson,J.D., C l a s s i c a l Electrodynamics , John Wiley and Sons, New York(1975) 5Friedmann,D.E., Curzon,F.L., Feeley,M., Young,J.F. and Auchinleck,G., Rev. S c i . Instrum. , Vol. 53, 1273(1982) -123 -APPENDIX A In chapter 2, section 3, I gave the moments of equation 2.3-19 with respect to Legendre polynomials. In t h i s appendix these quantities w i l l be derived. There are two cases to consider, the moment for the X-direction "switch on" problem and the moment for the Z-direction problem. For the X-direction problem the moment of the equation is taken with respect to P/vOl) and with respect to ^(71) for the Z-direction problem. A-1 - MOMENT OF EQUATION 2.3-19 FOR THE Z-DIRECTION PROBLEM For the Z-direction "switch on" problem, equation 2.3-19 becomes: 2.3-11, 2.3-24 and 2.3-26 respectively. As a reminder these equations are l i s t e d below: A. 1 - 1 The quantities , 5^ 2. and Ij. are defined in equations 2.3-11 2.3-24 -124 -Note, 97 has replaced 7?0 in these equations since t h i s point i s no longer fixed. Substitute these equations into equation A.1 - 1 to obtain: 2 R>^c^«^j^^ + 7 ^ ^ ^ ZnS&=iP A. 1-2 Multiply this equation by fjft??) and integrate over7): The second term in t h i s equation can be evaluated with the use of the identity shown below:3 Thus equation A.1-3 becomes: + j I^ WX A.1-5 Equation A.1-5 can be written in the form: -125 -A. 1-6 J -» A.1-7 A-2 MOMENT OF EQUATION 2.3-19 FOR THE X-DIRECTION PROBLEM For the X-direction "switch on" problem, equation 2.3-19 becomes: V f l x + ' X x k'XX - O A.2-1 The quantities b-\y. and I x are defined in equations 2.3-11, 2.3-25 and 2.3-27 respectively. As a reminder, equations 2.3-25 and 2.3-27 are: M € -1)"*- £ RJAKMCOS 4> 2.3-25 Note 7l„ and <£>0 are replaced by D and ty since t h i s point i s no longer fixed. Substitute equations 2.3-11, 2.3-25 and 2.3-27 into equation A.2-1 to obtain: -126 -2ja(tNift-Multiply t h i s equation by PJtfi) and integrate over?? : ZCdtt-i y ^ ' ^ V V ° ' ~ A.2-3 The second term in this equation can be calculated with the use of the identity shown below:3 I n j — I A.2-4 Hence equation A.2-4 becomes: n.&-ifHzH+\)iK Rudolf*) f p U j L ^ 1 _ ayfft "Ip'mj] Equation A.2-5 may be written: - 1 2 7 --I A . 2 - 7 -128 -APPENDIX B B-1 SOLUTION OF THE X-DIRECTION "SWITCH ON" PROBLEM FOR AN ASYMPTOTICALLY LONG SPHEROID In chapter 2, section 5, I obtained the equations which govern the response of a long conducting spheroid to an e l e c t r i c f i e l d applied perpendicular to i t s axis. In t h i s appendix these equations w i l l be solved. As a reminder these equations are l i s t e d below: -1 2.3-32 2.4-16 2.5-2 2.5-1 2.2-13 The procedure used to solve these equations i s outlined in th i s paragraph. F i r s t l y the dimension of matrix i s chosen , say N-dimensional, and the elements of E^itf are -129 -found. Next the e i g e n v a l u e s , ^ , can be found with equati 2.3-32: ion ^ [ B ^ A , + T ^ u ] = o B. 1 -1 The eigenvectors, l^yX, can be found to within undetermined c o e f f i c i e n t s with equation 2.3-32: B. 1-2 The undetermined c o e f f i c i e n t s are removed with the "equilibrium condition", equation 2.5-2. The eigensolutions, S/v"X(i*) , are then found with equation 2.5-1. F i n a l l y the response of the bulb i s found with equation 2.2-13. This problem may be s i m p l i f i e d considerably i f the symmetry of the problem to be solved i s considered. The function r^ u(7l) i s even w h e n i s odd and odd when^ Thus the matrix ByiN i s of the form: i s even. x o x o x O x o x o X O X o X o x o x o x o x where x i s a non-zero quantity. However, the equilibrium charge density d i s t r i b u t i o n , SffkiY*) , must be even to a change in the sign of 7) by the symmetry of the problem. Hence, from an inspection of 2.5-1, the eigenvectors, R^X , must be of the form: -130 -From an inspection of o^H and ry*> i t can be seen that the elements of B^ utf in which bothyiA and A/ are even play no role in determining the eigenvectors, R/A . Hence they w i l l not be included in the c a l c u l a t i o n . To s t a r t , evaluate B^KN . The l i m i t A —* O of Qi l^W) i s : Thus equation 2.4-6 becomes: -I Let: B. 1-3 B. 1-4 J | - 7 1 - U " B.1-5 -I Substitute equation B.1-5 into equation B.1-4 to obtain: 7*^  Za6 . (N t i )^AVL B - 1 - e Substitute equation 2.4-12 into equation B.1-6 to obtain: - 1 3 1 -D' XUZN-t-l) T N , N Substitute t h i s equation into equations B . 1 - 1 and B . 1 - 2 to obtain: The Jyjiw have been calculated for y u = 1 , 3 , 7 and N = 1 , 3 , 7 where the Legendre polynomials are: 3 P5'=/F^(ai,n+-I4-77N-|)£ P,»/rrrF(4Z? n *- *fi 5 13 s rt- 5 ) ^ The I ^ w a r e : I „ = 1 . 5 7 0 7 I „ = 0 . 5 8 9 0 I , 5 = 0 . 3 6 8 2 In = 0 . 2 6 8 4 I j 3 = 5 . 7 4 3 2 I , 5 = 2 . 5 5 4 1 \ n = 1 . 7 3 1 5 I55 = 1 0 . 6 5 6 I S 7 = 5 . 1 3 6 6 -132 -I 7 7 =1 6.035 where = Iw/u. . The eigenvalue problem has been solved for 1, 3, 5 and 7 dimensional matrices. These results are presented below. B.2 1-DIMENSIONAL EIGENVALUE PROBLEM With equations B.1-8 and B.1-9 and the [pm the one dimensional eigenvalue problem i s : B.2-1 B.2-2 Hence: -\ _ 0.589-TL A x~ R£ 0 B.2-3 R'/?s= X B.2-4 where x i s an undetermined c o e f f i c i e n t . Substitute equation B.2-4 into the "equilibrium condition", equation 2.5-2,to obtain: Hence: -133 -B.2-6 Substitute equation B.2-6 into equation 2.5-1 to obtain: B.2-7 Substitute equation B.2-7 into equation 2.2-13 to obtain: B.2-8 B.3 3-DIMENSIONAL EIGENVALUE PROBLEM Substitute the into equations B.1-8 and B.1-9 to obtain: [0,5890-0.08590 o.8376 - R ^ * * 0.5890 - E f j k * O.0 2 . Z 0 9 0 . 0 8 5 9 O 0 . 8 3 7 6 -The solutions of th i s eigenvalue problem are: Xx" °'^^ _ n' w i t h eigenvector R'^TMX** B.3-1 B.3-2 B.3-3 -134 -with eigenvector R^»AiX= y -O.GI78 -o.8fo6f B.3-4 where x and y are undetermined c o e f f i c i e n t s . Substitute equations B.3-3 and B.3-4 into equation 2.5-2 to obtain: [-0.<*636 R(77) - 0 - 6 1 7 8 R^7)| I" The solution of these equations i s : o . B.3-5 - 0 . 2 6 7 3 € j i x B.3-6 B.3-7 Substitute these equations into equations B.3-3 and B.3-4 to obtain: B.3-8 £°E X B.3-9 Substitute equations B.3-8 and B.3-9 into equation 2.5-1 to obtain: 0.834<J I-O.Z3I5 0.1651 c.2.316 o '^X.xVT)~ ^0^ 4(0.8^ 9 -0.3473 C57> - 1 ) ) B.3-10 -135 -B.3-11 F i n a l l y substitute these equations into equation 2.2-13 to obtain the response of the s h e l l : (0.S34^-0.3473C5^-'))(l-e R f e ° J + B.3-12 (0.I&5I to.3473(5?? x-l1)(l-^ J The method required to solve the 5 and 7 dimensional problems i s i d e n t i c a l to the method used to solve the 1 and 3 dimensional problems. Since nothing new i s demonstrated by solving these problems in d e t a i l , I w i l l simply present the re s u l t s . (.6*74-- .4140( 577*-1) t. j74-4Ul?7*-|1 )V| - e R £ * J +(.Jl ex - .2.9 10(577**-/) -.32.1 b (2,1T)4- \47th e "^£P] + (.0444- ,|iJO ( 5771- iH.(472-fej?7+-t477N-/|(| - e ^ R f e ? ^ B.3-13 & 7 XIV*, •£)/€. E* Co s £ ^ (.5093- .3753(577-l)4-.2554(2J Tr*-I477V 1) -.c£o«K^ 7j4-4*>S??* iiS7v-5))(i- C +(35o5r.087«S(^ ,--i)-.%o2{£ I 77*-|4?7 +•)+-0472(42.9?»6-4<?9fa\%stf-fy-of& \^ +(. 1 1-23 o 1 (S^- )-.o2.9M 2l77*- l47fH I ) - .0413 (4Z7 ?76-4 9S?7 + Hztf^jfy. B.3-14 -136 -APPENDIX C C.1 SOLUTION OF THE Z-DIRECTION "SWITCH ON" PROBLEM FOR AN ASYMPTOTICALLY LONG SPHEROID In chapter 2, section 6, I obtained the equations which govern the response of a long conducting spheroid to an e l e c t r i c f i e l d applied p a r a l l e l to i t s axis. In th i s appendix these equations w i l l be solved. As a reminder these equations are l i s t e d below: 2 RUXB£W •+• A z R W X = ° 2-3-28 2.4-13 Q,°Ci +A)(£A^ ^ > K ' U^l-yW^-t^) 2.2-,3 The procedure used to s o l v e these equat ions i s o u t l i n e d in t h i s paragraph . F i r s t l y the dimension of mat r i x E & y i s -137 -chosen, say N-dimensional, and the elements of BJLw are found. Next the eigenvalues, Az , can be found with equation 2.3-28: +'X zuJ = 0 c. 1-1 The eigenvectors, ry,x, can be found to within undetermined c o e f f i c i e n t s with equation 2.3-28: The undetermined c o e f f i c i e n t s are removed with the "equilibrium condition", equation 2.6-2. The eigensolutions, 8*A£(r), are then found with equation 2.6-1. F i n a l l y the response of the s h e l l i s found with equation 2.2-13. F i r s t l y I w i l l demonstrate that the integral in equation 2.4-13 i s equivalent to ~J^XN defined in the previous appendix. Let: 3 Substitute the following i d e n t i t y into equation C.1-3: C. 1-4 The result of the substitution i s : -138 -C. 1-5 w » j-— I Integrate t h i s equation by parts. Note, the polynomials v ^uCfl) are equal to zero when 7\ = ± | . Hence: -I Substitute equation C.1-4 into equation C.1-6 to obtain: C. 1-7 — I where the T^N have been calculated in the previous appendix. Substitute equation C.1-7 into equation 2.4-13 to obtain: This problem may be s i m p l i f i e d considerably i f the symmetry properties of the problem are considered. Since .^(T>) i s even whenyu. i s odd and f '^C )^ i s odd when^ i s even, matrix OJJ&I i s of the form: " x o x. o x o x o x o X O X O X O X O X o x o x o x where x i s a non-zero quantity. However the equilibrium -139 -charge density d i s t r i b u t i o n , S 2(.r ;i) , must be odd to a change in sign of 77 by the symmetry of the problem. Hence, from an inspection of 2.6-1, the eigenvectors, ryCX , must be of the form: X o X o From an inspection of B^, and iyuCk i t can be seen that the elements of in which b o t h a n d N are even play no role in determining the eigenvectors, fyuC< • Hence they w i l l not be included in the c a l c u l a t i o n . To s t a r t , evaluate the B^.A/ . The l i m i t s A—>0 of are: u Q;(I+A)R°(I t A ) _L 2 Substitute equations C.1-9 into equation C.1-8 to obtain: B,° ~ *(2.35 6 _ 4.7., 2_) B j , — If (o.8g30|n^) - 3 . 2 . 3 9 ) Bs, — y(o.5 5 2 i l « ( - Z.5 a 2.) B7<~y(o.4-DZ6 -2.088) C.1-9 -140 -Y(83*>7 |n(2^ -4-0.82.) B y ? ^ ( 6 , 0 6 0 m)-3l.4-$) g^' f rU.OZS Uj^j-4:050) B ^ - ^ g . ^ l In^-Zfo7.7) Bys—*(zo.Z6 1^-146.5) B,°7^y(2 toi3 ln(^- 4.0 Bj 7 ~YO 2.99 1*^-47.62) 857 - V(^8.5Z - 175.7) where V i s : Substitute equations 2.4-11 and 2.4-12 into equation C.1-10 to obtain: €.L> C.1-1 1 In t h i s problem the A dependence of the cannot be factored out. Hence the eigenvalue problem must by solved for each chosen A. This problem w i l l be solved forA. ranging from IO to lO . F i r s t l y , l e t A=.o2_. Substitute A=o.o2, into the B > * N to obtain: -141 -0.8 2 7 B°j 5 ^ ' V 13.183 0 B°7/~ tfb.2.34 B ^ - Y 18.86 Bs-r^-y I2..I90 -YO .344- Bg-,A/ - y i A 8 ^ B 7 7^*69.8l8 C.2 1-DIMENSIONAL EIGENVALUE PROBLEM Substitute the ^uw into equations C.1-1 and C.1-2 to obtain: cJet[fe.l36- ^ " ] = 0 C.2-1 [6.138 - ^ ] R ^ 7 C ° C.2-2 Hence: ^ = V 6.1383 C.2-3 C.2-4 where x i s an undetermined c o e f f i c i e n t . Substitute equation C.2-4 into the "equilibrium condition", equation 2.6-2, to obtain: -142 -/ " V XR°(?7) C.2-5 Hence: Substitute equation C.2-6 into equation 2.6-1 to obtain the eigensolution, S/^C^) : Substitute equation C.2-7 into equation 2.2-13 to obtain: C.3 3-DIMENSIONAL EIGENVALUE PROBLEM Substitute the into equations C.1-1 and C.1-2 to obtain: 4..I38 - 5.3*72-Jet = 0 c.3-1 0.62.^3 18.87- ^ '6.138 - ?f- 5372. 0.6 Z73 I 8 . B 7 - ^ | ^ / A X - 0 C.3-2 -143 -The solutions of th i s eigenvalue problem are: with eigenvector R^,* X Q.997 -O.063 C.3-3 with eigenvector Ryu7vzl=y 0.380Z C.3-4 where x and y are undetermined c o e f f i c i e n t s . Substitute equations C.3-3 and C.3-4 into the "equilibrium condition", equation 2.6-2, to obtain: 'oSftl R°(7>) 0 . 3 8 0 2 . P>)' O -0.O63P°0?) 0 . 9 4 - 9 Q,°(I+A)(2-AY4. The solution of these equations i s : X Lyj c.3-5 y QJC 1 +A) C2A^ C.3-6 C.3-7 Substitute these equations into equations C.3-3 and C.3-4 to obtain: O.T75 -0.062. C.3-8 -144 -O.OT5 O.ObO £*Ez C.3-9 Substitute equations C.3-8 and C.3-9 into equation 2.6-1 to obtain: C.3-10 C.3-11 Thus, from equation 2.2-13, the response of the s h e l l i s : W^^ ^75 -0.oS.C577^-3)Xl-e^t) _ t9.2|JCLD\ C. 3-1 1 ^zQX '+^KaA^ypTry^Ea.?? i s plotted against n.Dt/Ll£o for various 71 in figure 43. Furthermore: _ 5.79.n.Dt I - Q C.3-12 is plotted on the same graph. From an inspection of the graph i t i s apparent that 85 ZQ,°(| + ^ R^VVl-? 1 V^Eiz.7i i s e s s e n t i a l l y independent of 7) . Moreover i t may be approximated by equation C.3-12 since the experiments I w i l l perform could not possibly detect the error resulting from - 1 4 5 -FIGURE 43 -146 -th i s approximation. Note, I would expect the time constant to be independent of 71 from the geometry of the problem. Thus, assume: The method used to solve the 5 and 7 dimensional problems is i d e n t i c a l to the method used to solve the 3 dimensional problem. Since nothing new i s demonstrated by solving these problems in d e t a i l , I w i l l simply present the r e s u l t s . It turns out that S^ Cr^ -t-) and S7zO^ t) are equal to S^Cv^t-) for the number of s i g n i f i c a n t figures considered. Hence: C.3-14 Thus: This problem was solved for various A using the same technique outlined above. The results are: -147 -EMPIRICAL F I T OF X < A 2 c 0L 2/DA FIGURE 44 -148 -'Xa£oL"/DJL i s plotted against A in figure 44. These results l i e on a straight l i n e which may be approximated by: - f j ^ - 2 . - 2 6 ln(3.82vA) C.3-16 as demonstrated in figure 44. Substitute equation 2.4-11 into equation C.3-16 to obtain: C.3-17 Hence the response of the s h e l l may be written: where: U ~ I h ( W I ^ ) C.3-19 -149 -APPENDIX D D.I SOLUTION OF THE X-DIRECTION "SWITCH ON" PROBLEM FOR A SPHEROID TAKEN TO THE SPHERICAL LIMIT In chapter 2, section 7, I obtained the equations which govern the response of a spheroidal s h e l l , taken to the spherical l i m i t , to an applied f i e l d in the x-direction. In th i s appendix these equations w i l l be solved. As a reminder these equations are: 2.7-5 36.ExP/(nw££R^P ,(7i) S/\X '">~ C / i \ i i f t i , . i / i n osu; 2.7-6 ) ^ ~ ^ r y A ^ ( 7 l ) C o 5 < t > 2.2-13 The procedure used to solve t h i s problem i s outlined in th i s paragraph. F i r s t l y the elements of B^n are found. It turns -150 -out the matrix i s diagonal. Thus, from equation 2.7-4, the eigenvalues are: D. 1-1 and the eigenvectors are: I O O o I o O = 2 O O I o where x, y, z, etcetera are undetermined c o e f f i c i e n t s . The eigenvalue -&',, corresponds to the eigenvector with c o e f f i c i e n t x, ~B>ZJL to the eigenvector with c o e f f i c i e n t y, etcetera. An inspection of the "equilibrium condition", equation 2.7-8, indicates that and a l l other undetermined c o e f f i c i e n t s are equal to zero. Hence the eigenvector i s : o o o D. 1-2 Thus, from equation 2.7-6, the eigensolution i s : D. 1-3 From equation 2.2-13, the response of the s h e l l i s : -151 -D. 1-4 A l l that i s l e f t to do i s to find the elements of B^»/ Let: D. 1-5 This integral can be evaluated a n a l y t i c a l l y with the use of the following i d e n t i t i e s : ^ C 2 y x + i ) 7 l f > ' ( r j ) ' / ^ . ( ^ ) H/^-0^,(7)) D. 1-6 D. 1-7 Substitute equation D.1-7 into equation D.1-5 to obtain: D. 1-8 Substitute equation D.1-7 into equation D.1-8 to obtain: d?7 = 0 D. 1-9 Substitute equation D.1-6 into equation D.1-9 to obtain: -152 --I Thus: P>)^ 'Cr))(^ f.^ c)77 D. 1-10 D. 1 -11 This integral can be evaluated with the use of the following identity:^ I R.'m^nlcJ'n = ^ - ^ ( ^ + 0 ^ D.1-12 —i Hence: K^dak2j&mi D.,-,3 The l i m i t of R'(!JQ,'Cf„) i s : ? —•• oo J S a Hence B,' becomes: The radius of the sphere, from equation 2.7-3, i s : -153 -R ^ % ^ D.1-16 Substitute equation D.1-16 into equation D.1-15 to obtain: ^ ' ^ ^ f t " D. 1 - 1 7 From equation D.1-4 the response of the s h e l l i s : 6xCf\t) ^3£„R'(7))Co4(| - D.i-18 where: -154 -APPENDIX E E.1 SOLUTION OF THE Z-DIRECTION "SWITCH ON" PROBLEM FOR A SPHEROID TAKEN TO THE SPHERICAL LIMIT In chapter 2, section 8, I obtained the equations which govern the response of a spheroidal s h e l l , taken to the spherical l i m i t , to an applied f i e l d in the Z-direction. In th i s appendix these equations w i l l be solved. As a reminder these equations are: 2.8-2 . 2.8-1 2.8-5 2.8-3 2.2-13 7\ The procedure used to solve t h i s problem i s outlined in t h i s paragraph. F i r s t l y the ' are found. It turns out the matrix i s diagonal. Thus, from equation 2.8-2, the -155 -eigenvalues are: and the eigenvectors are: E. 1-1 i o o o o I 0 o o o where x, y, z, etcetera are undetermined c o e f f i c i e n t s . The eigenvalue -B t l corresponds to the eigenvector with c o e f f i c i e n t x, " B ^ to the eigenvector with c o e f f i c i e n t y, etcetera. An inspection of the "equilibrium condition",, equation 2.8-5, indicates that x= Se.o£-£%Q and a l l other undetermined c o e f f i c i e n t s are equal to zero. Hence the eigenvector i s : f^xi ^ 3e .E £ f o ! o o o E. 1-2 Thus, from equation 2.8-3, the eigensolution i s : E. 1-3 From equation 2.2-13, the response of the s h e l l i s : -156 -E. 1-4 A l l that i s l e f t to do i s calculate the elements of Bp* • Let: E. 1-5 This integral can be evaluated a n a l y t i c a l l y with the use of the following i d e n t i t i e s : 3 E. 1-6 E. 1-7 Substitute equation E.1-7 into equation E.1-5 to obtain: I E. 1-8 Substitute equation E.1-7 into equation E.1-8 to obtain: I E. 1-9 Substitute equation E.1-6 into equation E.1-9 to obtain: -157 -E. 1-10 - 1 Thus: - l This integral may be evaluated with the use of the following i d e n t i t y : E. 1-12 E. 1-13 -1 Hence: The l i m i t of ^iS . ^ t f J ) i s : U PfltoQ^.)~-^ E.1-14 Hence B,', becomes: The radius of the sphere, from equation 2.7-3, i s : E. 1-15 -158 -Hence: From equation E.1-4 the response of the s h e l l where: 

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