UBC Theses and Dissertations

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Bipolar induction torque in a circular disc Morton, Ralph Mackenzie 1933

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IBRARY I  u I C A T . MO.  S  •~  BIPOLAR IliDUCTIGF TORQUE . IE A CIRCULAR DISC  R a l p h IvIaeKenzie Morton.  A T h e s i s s u b m i t t e d f o r t h e Degree o f MASTER OF APPLIED SCIENCE i n the. Department of ELEC TRICAL E1GI1EERI1G  The U n i v e r s i t y o f B r i t i s h JU?RIL 1.933  Columbia.  k^L^m^M^Lk^  i  TABLE OF CGUTEITTS  Page Introduction -  1  I n v e s t i g a t i o n o f Eddy Current Path C a l c u l a t i o n ' o f Torque A p p e n d i x Index  2 7 13  B i p o l a r Induct I o n Torque i n a C i r c u l a r D i s c .  I n order t o compute the t o r q u e of an i n d u c t i o n i n s t r u m e n t I t i s n e c e s s a r y t o know the magnitude and d i r e c t i o n of the c u r r e n t s i n d u c e d i n the moving element. I n d e v e l o p i n g the t h e o r y o f the t o r q u e due  to  bipolar induction i n a f l a t circular disc, i t i s usually assumed t h a t the eddy c u r r e n t s f l o w a r o u n d the i n d u c i n g magnet i n a s e r i e s of c o n c e n t r i c r i n g s . While t h i s a s s u m p t i o n i s v a l i d , i n the case where the c e n t r e o f the magnet c o i n c i d e s w i t h the c e n t r e of the d i s c , i t I s not so when the magnet i s l o c a t e d e c c e n t r i c a l l y w i t h r e s p e c t t o the d i s c a x i s ; and torque f o r m u l a s based on such an a s s u m p t i o n g i v e r e s u l t s c o n s i d e r a b l y i n e r r o r , the e r r o r i n c r e a s i n g as the magnets approach the edge o f the disc. D r y s d a l e and J o i l e y  s t a t e t h a t the  theoretical  torque i s about 22 0% o f the measured t o r q u e , the  ratio  v a r y i n g w i t h the d i s t a n c e between the magnets and the  disc  edge.  D r y s d a l e and J o l l e y , E l e c t r i c a l M e a s u r i n g I n s t r u m e n t s , Y o l * 2, P. 168. E r n e s t Benn, L t d . , London.  The o b j e c t of t h i s paper i s t o i n v e s t i g a t e the p a t h of the eddy c u r r e n t s i n d u c e d i n a f l a t c i r c u l a r d i s c , and t o d e v e l o p a trip que f o r m u l a based on t h e a c t u a l c u r r e n l distribution. I n v e s t i g a t i o n o f Eddy C u r r e n t P a t h . C o n s i d e r an a l t e r n a t i n g c u r r e n t f i e l d normal t o an i n f i n i t e  sheet o f c o n d u c t i n g m a t e r i a l as i n F i g . 1.  The r e s i s t a n c e o f an a n n u l a r r i n g o f mean r a d i u s R and w i d t h dR i s g i v e n by t where and  <5~ t,  i s the s p e c i f i c r e s i s t a n c e o f t h e m a t e r i a l  the t h i c k n e s s o f the sheet. 'The conductance of the r i n g I s g i v e n by 2T1 <S~~ Since the  -7?  <f^/7?.y~. i n d u c e d i n a l l r i n g s i s the  same, the c u r r e n t f l o w i n g i n a r i n g of o u t e r r a d i u s Rg and inner radius R  (  is  2'77<T e y  • Tf,  Mext c o n s i d e r a p a i r of e q u a l a l t e r n a t i n g c u r r e n t f i e l d s o f o p p o s i t e p o l a r i t y normal t o an i n f i n i t e sheet as shown i n F i g . 2 .  conducting  Let  P  move i n the d i r e c t i o n o f the r e s u l t a n t  e l e c t r i c f i e l d , and l e t F Since  he on the l o c u s o f P.  P P i s the l o c u s o f P,  t h e component o f  the e l e c t r i c f i e l d n o r m a l t o PI? must be z e r o .  That i s t o  say t h e r e can be no. r e s u l t a n t c u r r e n t c r o s s i n g t h e l o c u s  PP.  The c u r r e n t c r o s s i n g PF i n d u c e d by the p o l e a t JL I  o  . =  Et  log  i  R  The c u r r e n t c r o s s i n g FF i n d u c e d by t h e pole a t B ! '  =  E_t_ l o g £2 j?77(f ^ IT  _  The r e s u l t a n t c u r r e n t log  RL  f_  L O  m  I  a  + 1^  •%  .  must be zero .0  , •  Next f i n d t h e e q u a t i o n o f t h e l o c u s o f P. to F i g . 3• Take t h e o r i g i n a t 0. 2 > ,2 2 R = (s +x; *y = 2 g 2 Rg = ( s - x ) +y = /  _ 2 [ V / R *• 2 ?, 2 o 2 Z, E ( s -2sx+x +y ; But  . 2o * y r  f r  x -  _  2 2 2 s +2sx+x +y p 2 2 a —2sx+x +y  2  Z, 2 2 2 = n ( s +2sx+x +y ) 2 TI 2 2. 2 m2 *+n' n, 1 mmn n I  m2 c  2 1  2(m-n)J  2  2  2  (m-n)  2  ^  ,  Refer  _4-  T h i s i s the e q u a t i o n o f a f a m i l y o f c i r c l e s of radius  ...fti* ... m-n  having- t h e i r c e n t r e s a t 2 2 m -MI ^bi-n) y = 0  1  Equation  ( l ) d e s c r i b e s the p a t h o f the eddy  c u r r e n t s i n d u c e d i n a t h i n i n f i n i t e sheet by a p a i r o f e q u a l a l t e r n a t i n g c u r r e n t p o l e s l 8 0 ° o u t o f phase w i t h each  other,  A s e r i e s of these c i r c l e s t o g e t h e r w i t h a s k e t c h o f t h e p h y s i c a l arrangement o f the p o l e s i s shown i n F i g . 4 and 5 . I n o r d e r t o d e s c r i b e the f l o w c i r c l e s i t i s conv e n i e n t to,know t h e r e l a t i o n between s  =  the h a l f d i s t a n c e between the p o l e c e n t r e s .  R  =  the r a d i u s o f any c u r r e n t path.,  •3,  -  the d i s t a n c e between the c e n t r e o f the c i r c l e o f r a d i u s R and the c e n t r e o f t h e i n d u c i n g pole.  T h i s r e l a t i o n i s o b t a i n e d as f o l l o w s : s _  m^+n^  _  m+n  2(m-n)  .  2  n .  =  2  m-n s  =  m+n .2'  .".  m  ~  2s - X  n  =  R -  =  .(R  e  e  ,2  - e ) '  2s - R + <S - R +  R  2  -  2es  S  =  - €  R  2  =/s'  2  = ; . 0  ~G  (2)  C  2e 2  + R  2  - S:  . (3)  I t .should be n o t e d from (2) t h a t  8. i s the  parameter, clef i n i n g ; an e n t i r e family,,of c u r r e n t ; f l o w c i r c l e s , , and t h a t i f t h e r a d i u s o f one c i r c l e ;i's 'known"'together w i t h the d i s t a n c e between i t s c e n t r e and t h e c e n t r e of t h e p o l e , then of of  S S  c a n be computed; and b y s u b s t i t u t i n g t h i s .value i n (j)» t h e v a l u e o f R  corresponding' t o any v a l u e  can'be f o u n d , thus d e f i n i n g the e n t i r e f a m i l y o f  flow.lines'for.'that p a r t i c u l a r  case,  A s e r i e s o f f l o w l i n e s drawn b y the above method i s shown i n F i g . 6. Consider  an a l t e r n a t i n g ; c u r r e n t p o l e l o c a t e d  e c c e n t r i c a l l y w i t h r e s p e c t t o a conducting- d i s c . Let. R' -  = r a d i u s of d i s c  t •'•.••<£? -  d i s t a n c e between c e n t r e o f p o l e and centre of d i s c .  Then the boundary o f t h e d i s c de f i n e s . one - f l o w l i n e , and by means o f e q u a t i o n s .(2) and ( 3 ) the', f l o w l i n e s w i t h i n the d i s c can be computed. •  I n order t o o b t a i n some.experimental check on  the p r e c e d i n g work, a s h a l l o w c i r c u l a r t r o u g h o f mercury was s e t up i n c o n j u n c t i o n w i t h a n a l t e r n a t i n g , c u r r e n t magnet  as shown i n F i g . 7*  Two p o i n t e d e l e c t r o d e s made c o n t a c t w i t h  the mercury, and were connected t o a v i b r a t i o n galvanometer t u n e d t o the s u p p l y f r e q u e n c y , p r o v i s i o n b e i n g made f o r moving these c o n t a c t s and p l o t t i n g t h e i r p o s i t i o n r e l a t i v e t o the d i s c . The  e q u i p o t e n t i a l l i n e s i n the mercury d i s c were  f o u n d and p l o t t e d by means o f t h i s a p p a r a t u s .  The method  gave good r e s u l t s , because a v e r y s l i g h t d e v i a t i o n o f one e l e c t r o d e f r o m the e q u i p o t e n t i a l l i n e p a s s i n g t h r o u g h t h e second, e l e c t r o d e caused a l a r g e d e f l e c t i o n o f t h e g a l v a n o meter.  A p l o t o f the e q u i p o t e n t i a l l i n e s to f u l l s c a l e i s  shown i n F i g . 6. The  e q u a t i o n o f the e q u i p o t e n t i a l l o c i i s t h e  orthogonal t r a j e c t o r y of the o r i g i n a l flow l i n e equation ( l ) and i s g i v e n by  .  a , _ > ,2 , / 3c + i y +• o) = v s  2  + b  2  This defines a f a m i l y of c i r c l e s having  their  c e n t r e s on t h e y - a x i s and p a s s i n g t h r o u g h t h e c e n t r e o f each o f the p o l e s . By means o f e q u a t i o n s (2) and (3) t h e y - a x i s was l o c a t e d f o r the value of R  and  u s e d i n t h e mercury s e t  up, and t h e s e r i e s o f e q u i p o t e n t i a l and f l o w c i r c l e s was drawn. The  e q u i p o t e n t i a l c i r c l e s d e s c r i b e d by the above  method were found t o c o r r e s p o n d v e r y w e l l w i t h t h e e x p e r i -  m e n t a l r e s u l t s , and c o n s t i t u t e d a check on t h e p r e c e d i n g theory.  I n F i g . 6, the t h e o r e t i c a l e q u i p o t e n t i a l c u r v e s  are drawn i n f u l l , w h i l e the e x p e r i m e n t a l p o i n t s are shown by t h e s m a l l c i r c l e s . C a l c u l a t i o n o f Torque, The p r e c e d i n g work i n d i c a t e s t h e p a t h o f the c u r r e n t s i n d u c e d i n a, d i s c .  The next step i s t o develop  torque r e l a t i o n s based on t h i s c u r r e n t d i s t r i b u t i o n . I n o r d e r t o compute the t o r q u e produced by the f l o w o f c u r r e n t i n d u c e d i n the d i s c by one magnet a c r o s s the f i e l d o f the second, magnet, i t i s n e c e s s a r y t o know t h e magnitude and d i r e c t i o n o f the c u r r e n t . The following d a t a i s r e q u i r e d : Diameter of d i s c . Thickness o f d i s c . Specific resistance of disc material. Dimensions o f magnets. L e n g t h o f a i r gap. Location of poles. Magnitude and phase of f l u x i n each magnet. Supply f r e q u e n c y . R e f e r t o F i g . 8. The f i r s t c i r c l e s through under t h e p o l e the p o l e  A,  g B  s t e p i s t o d e s c r i b e the f l o w l i n e and  h.  Then a l l t h e c u r r e n t t h a t f l o w s  due t o the Cmf*  i n d u c e d i n t h e d i s c by  must f l o w i n the c r e s c e n t bounded by these  two c i r c l e s . Construction:  Given.  Evaluate  s  &  -  Ro  =  uv  d i s t a n c e between c e n t r e . o f d i s c and c e n t r e o f p o l e A. radius; o f d i s c .  by means o f e q u a t i o n  Draw t h e y - a x i s a t a d i s t a n c e pole  (2).  s from t h e c e n t r e o f  A. Bisect  uh  at K  and draw kh  s e c t i n g t h e y - a x i s a t b.  Join  bh  normal t o uh i n t e r and draw  hb  normal  to hb  and i n t e r s e c t i n g t h e x - a x i s a t b . W i t h c e n t r e b t and r a d i u s b h d e s c r i b e a c i r c l e . T h i s c i r c l e i s t h e f l o w l i n e through the p o i n t  h.  By means o f t h i s c o n s t r u c t i o n draw f l o w c i r c l e s through  g  and  j , the d i a g o n a l l y o p p o s i t e c o r n e r , and  centre of the pole, r e s p e c t i v e l y . Let  R,  = r a d i u s of f l o w c i r c l e through  g.  Rg  = r a d i u s of f l o w c i r c l e t h r o u g h  h.  & i d* '  2  d i s t a n c e between c e n t r e o f R . c i r c l e and c e n t r e of p o l e A. = d i s t a n c e betT/een c e n t r e o f Rg c i r c l e and c e n t r e o f p o l e A.  T  = t h i c k n e s s of d i s c .  6~  •- s p e c i f i c r e s i s t a n c e microhms/cm^. / 2 2 = / S * R, ' - S  ^  =/s  2  + R  2 2  _ . S  A l l dimensions i n centimetres.  ../Knowing t h e above s i x f a c t o r s i t i s p o s s i b l e to work out t h e r e s i s t a n c e o f t h e r i n g .  The c a l c u l a t i o n i s  r a t h e r l o n g and'Is . g i v e n i n t h e appendix. The f i n a l , e x p r e s s i o n i s ~  Resistance  277 0"  _  -6  -r X  :  ohms a  10  log  • e, 2  e  1  R  The current, f l o w i n g i n t h e r i n g can now be computed. Let  0 j •. *  f l u x i n magnet  y~  supply  -  -  A,  Tptf-S.  frequency..  .,- 'enif i n d u c e d i n r i n g by (j>  .  •'It,  -  r e s i s t a n c e of r i n g magnet A.'  I  =  c u r r e n t f l o w i n g i n r i n g due to E . a  ~  f l u x d e n s i t y o f magnet  • Q  =  e l e c t r i c a l phase angle between  L  -  mean l e n g t h o f p a t h o f c u r r e n t under magnet B. •  i  t (3  Then  E ^  B  '•«?•' 2 77 / <j> #/ o >  fk R  /  t  ^  i s obtained  =  surrounding  B.  g  ^ Z T l o g  e  ^ ^ ~ X  . 2 77 6" f r o m . E i g . . a s follows:  10  -10*  L e i . a ' , be the" c e n t r e of. the flow; c i r c l e t h r o u g h J . • (the c e n t r e o f t h e pole; ..B s e c t i n g the e i ^ Then  E  through  a n d Eg  R  at d  '^J draw and  s  ^^interrespectively,  L : may be t a k e n as t h e e f f e c t i v e a r e a / o f t h e p o l e  d i v i d e d ; by: .de, The  V:  .  .r  (  ;  M  /  i  ^  .  :  ^  ^  B ;  •-•  f o r c e , between the c u r r e n t induced' by /magnet A-  and the : f i e l d o f magnet->B> .ls""  " •'.'  ;  -v  .'  S i n c e the f l o w l i n e s ; under t h e p o l e  B  are n o t  p a r a l l e l t h e d i r e c t i o n o f the f o r c e Var.Ies over ^he p o l e . :  f a c e , f o r example, -the. f o r c e due t o the .current element a t .'the. p'oint.v- h)-; '-Would; be' .along hb' .and the; f o r c e due t o . the c u r r e n t 'at ;.g- would, be a l o n g  go',/  I t i s ' assumed  t h a t the r e s u l t a n t f o r c e a c t s along- Ja' , t h a t ; is. t o s a y , a l o n g t h e n o r m a l t o the f l o w l i n e p a s s i n g under t h e centre;: o f t h e pole.;' •;'.;."• ;  :  T h i s , force , can be r e s o l v e d . i n t o two components, :  one a c t i n g t h r o u g h the a x i s o f t h e d i s c , a n d the, .other tangential t o the disc.  The t o r q u e ,1s due t o t h i s second  component. '.Let' ^  =  angle . Ct'j.lTr.:;  _;;;-='radius, •;.\'...;Then''torque.',-.^".'. F  arm 7 V~ :  sin ^ >  /  .  '  •: .  -11-  • Since''an e q m l torque i s produced by the c u r r e n t I  s  f l o w i n g under, t h e f i e l d  The t o t a l t o r q u e i s t w i c e  the above, and c a n be e x p r e s s e d by T  =  0.2  x  2  1  1  f  x/^  B  i c / fsin &  sin  (Jj  10*  dyne cm. I n o r d e r t o check t h e p r e c e d i n g t h e o r y , an aluminum d i s c mounted on a v e r t i c a l a x i s I n j e w e l b e a r i n g s was;.'.fitted w i t h a; t o r s i o n head as shown i n F i g . 11. The :  i n s t r u m e n t was . c a r e f u l l y c a l i b r a t e d , and the t o r q u e developed by the d i s c under t h e i n f l u e n c e  o f two magnets connected t o  a two phase s u p p l y -was measured.  The f l u x from each magnet  was measured b y means o f s e a r c h c o i l s wound on t h e p o l e s and c o n n e c t e d t o an. AC p o t e n t i o m e t e r . The dimensions of t h e a p p a r a t u s were as f o l l o w s : Diameter o f d i s c 12*7 em, : .Thickness o f d i s c .IO55 cm. Specific resistance 2.83 microhms p e r cm. cube. . (aluminum) Dynamometer c o n s t a n t 1 8 . 5 dyne em. p e r degree. P o l e dimensions 2.2 x 1.9 cm. Search c o i l 20 t u r n s . The r e s u l t s o b t a i n e d w i t h t h i s a p p a r a t u s a r e g i v e n i n t h e ..appendix. p o l e s was  When ••'the.-space a n g l e between the ,  t h e c a l c u l a t e d t o r q u e v a r i e d from 63. t o 937°  of t h e measured t o r q u e , depending on t h e d i s t a n c e between the magnets and t h e edge o f the d i s c .  When t h e p o l e s were  spaced out t o 90° the c a l c u l a t e d t o r q u e v a r i e d from 5.6.5 t o 76.3fo o f the measured t o r q u e .  I t s h o u l d be n o t e d t h a t a number o f assumptions s i m p l i f i c a t i o n s have been made i n d e v e l o p i n g the p r e ceding theory.  I t i s d i f f i c u l t t o f i n d the magnitude and  d i r e c t i o n o f the r e s u l t a n t f o r c e due t o the c u r r e n t f l o w i n g under'the magnet, because of the v a r y i n g c u r r e n t d e n s i t y . The e f f e c t s o f the d i s c leakage i n d u c t a n c e on the f l o w c o n t o u r s has not been taken i n t o a c c o u n t .  I n the ease  of a low r e s i s t a n c e d i s c , the d i s c r e a c t a n c e would  decrease  the magnitude and change the phase angle of the induced currents.  The  \  t o r q u e would a l s o be m o d i f i e d by the e f f e c t  \ \  \  o f f r i n g i n g i n the d r i v i n g magnets.  \  I n view o f t h e number of v a r i a b l e s and the com-  \  \  v  p l e x i t y o f the phenomena i n v o l v e d , i t i s f e l t t h a t t h e t h e o r e t i c a l r e s u l t s are i n about as good agreement w i t h the o b s e r v e d q u a n t i t i e s as c o u l d be e x p e c t e d . A p p a r e n t l y v e r y l i t t l e work has been done on the s u b j e c t , and i t i s p r o b a b l e t h a t f u r t h e r r e s e a r c h w o u l d • r e s u l t i n the d e t e r m i n a t i o n o f an a c c u r a t e e x p r e s s i o n f o r the torque..  i y /  /  -13  ..APPENDIX. Table of r e s u l t s . Plotted  results.  Worked, o u t Example. Calculation Drawings.  of Resistance.  -14-  P o l e A n g l e , b5° D i s t a n c e 'between edge of p o l e and D i s c Edge. mm.  Measured Torque Dyne Cm.  Computed Torque Dyne Gnu  %  2260  1420  63  3710  2840  .. ' 7b.5  12  4?60  4100  86.0  17  5300  4920  93*0  1680  950  3b.3  3080  1950  63.  12  4050  3020 •  74.3  17  4550  3440  76.5  •  6  . - P o l e '..Angle. 0  -Id-  l e s t ITo. 13. 65 .  A n g l e between p o l e s  0  6m.m.  D i s t a n c e between edge o f p o l e and edge o f d i s c E.M.F. i n d u c e d i n 20 t u r n s e a r c h c o i l A n g l e between E  and E  a  =  D  Dynamometer r e a d i n g See page  jj  82.5  =  1.220 1.370  Pole A Pole B sinG  0  =  V ¥  .991.  201°.  =  f o r instrument  data.  F i r s t a p l a n of the d i s c was drawn showing t h e two magnets i n t h e i r p r o p e r l o c a t i o n as i n F i g . 8. Then •  S  was computed  ' R - e = _o 2  S  6.35  _ -  D  2  -  4.63  2 2.01.  =  2 x 4.55  By means o f the c o n s t r u c t i o n g i v e n on page f l o w c i r c l e s were d e s c r i b e d t h r o u g h t h e p o i n t s Measure  R R  e  » / s <£>  2  + R  B  =. 2.92  cm.  5.50  cm.  =  3  =  - S / " 2 // 2.01 2.01  =  / 2.01  JL  <f=  A  Resistance of r i n g  2  I  ~  2  + 2.92 2,  -i-  2  - 2.01  =  -2.01  =  - — 5 = — _  e  !  +'5.50  -AMI  X  T  2  x  _ I l o g 1.34  h, and j  I.33 3.87.  R  2.303 l o g . . _ J _ A . • ^ l B 1  2 77 x 2.83 .105  g,  =  0  H  574' microhms.  -lb-  Current f l o w i n g i n r i n g 1 0 . ss 10.62 amps. ~ ,1.220 -±u_ l  i  n  x  574  20  F l u x d e n s i t y i n magnet:  /3  %_  B  ,  x - i  V  2  pole area  «370 40 x 60 1  The  B  K  _iiL 4  e  =  432  lines  /  cm.  2  f o l l o w i n g q u a n t i t i e s a r e measured from F i g . 8, ^ajw  »  ^  =  25.5°  de  sin^  2.7  =  .430  =  em.  Mean l e n g t h o f p a t h under p o l e = = Torque  4  * 2.7 2  =  I.56  Radius.arm  =  =  -ft ^^.— 0  r  e  cm, 4 . 6 5 cm.  *• 0,2 x 10*62 x 452 ±'1.56-3: 4.65 x .991 x .430 =  Measured torque  =  2850 dyne cm.  2 01 x 18.5  =  3710  dyne cm.  Computed', torque- as p e r c e n t o f measured t o r q u e  -  76.5?° •  -17-  : C a l c u l a t i o n o f Ring. R e s i s t a n c e . Refer to F i g . 9 . Let  Rj  =  r a d i u s o f outer boundary o f r i n g .  R  =  r a d i u s of innter.boundary  0  of ring.  •^y.;^" d i s t a n c e between'magnet c e n t r e and centre of c i r c l e R . •  <?  2  =  d i s t a n c e betweeh magnet c e n t r e and c e n t r e o f c i r c l e R ••  ^'f.-'• \•>"''-di^.'- ='•;' s p e c i f i c r e s i s t a n c e 'of d i s c . m a t e r i a l . :1  - .-thiclcness o f disc'.  ,  •.;; T h e n p r o v i d e d X t h a t R^, i s o n l y s l i g h t l y t h a n Rg, t h e r i n g r e s i s t a n c e i s g i v e n by  m  I  2 2 l/2 ( a +R. + 2 a R c o s 6) ' - R  ' 0  '  '  'This i n t e g r a l i s e v a l u a t e d as f o l l o w s :  T  /  Let I = JJA^zl  f*j* ^*W ™*±J>* Z  x  / 2 2 2 ^ a -i-Rj - R +2aR cos © ^ R , <-R ) 2 /* ^ d© 5 / ~ g 2 ~ / a. +R -R -;-2aR cos O T  1  0  I  2  (  R  2  (  2  j  greater  E v a l u a t i o n of I Let U  (  -  t a n ;"• © • ' - • 2 .• • :  Then s i n 0  =  2U  1 -  u = e  = 77  2  / 0  Let When  U  tan =  ^  0  -2  •  |(a-R, } - R  U  =  1 + (/  S  0  OT^glj r  2 du  d0  ^  A  1  0  2)2 2  2du 2  >/•  1 *U  2  J yfc | (a+R, ) -R  2w g  2du  |/l+U  f 0  du  =  a * R, a -  sec  R.  2 17.  2d"  (R  ^)  |  (a+R,  ) see f  ^ >  h*> -\M) f  d ( t a n (y ) tan  see <f Z  *~V  J<f  > - 4 sec z  -19-  2^ =  (a+R,) - R 2  -  _- -£" 2  7 7 (T  x  U^R )  (a€,/  2  ^> )  <  R  2  a + R  i>  E v a l u a t i o n of I  j  =  <TR (R +R) / 2  (  J  1  2  -  2  a  2  .  + R,*  - ^  2 2  +  2aR  y  cos  o  Let  tan ^ Then ' cos 0  I  = 2  U  <T 2  =  •• s  R  T  9^  =  I  1-C/  2  1 *6>  2  -—~-—5-  /"  d  * >, y  ~  2a o 1 v(J  2  2du  2 2 _ 2 . 2._  / i-t^ »V I f  2  ^ /  0  20  2  7  ( ( a ^ j ^ j u ' ^ a ^ )  2^R (V- 2)  (a  +R )  2  1  «  I  +  I  2  •  s  - ^  f.ar,-V/ " « " 2  R  2  7? ( f R ( R  2  :  B  )2  R  }  2  2  . , '.  ' 77<T(R tR ) f  2  (R ^R ta) t  2  The above e x p r e s s i o n g i v e s the r e s i s t a n c e o f a c r e s c e n t provided, t h a t R, i s o n l y s l i g h t l y g r e a t e r t h a n  R^»  In order t o f i n d the r e s i s t a n c e o f a t h i c k crescent such as i n F i g . 1 0 , i t i s n e c e s s a r y t o change the above e x p r e s s i o n i n t o a conductance, and i n t e g r a t e t h e conductances between the o u t e r and i n n e r r a d i i o f t h e t h i c k c r e s c e n t . T h i s i s done as f o l l o w s :  let  c  =  conductance  R  2  =  R  R  }  =  R + dR  = —™-  -21-  -22-  = 2<f77 1  s  e  -  b  a  log  7/<f .  e  e  A a  2 Resistance  b  77 <T  — — x  —  ——  2.303 l o g  10  Fig 4.  

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