UBC Theses and Dissertations

UBC Theses Logo

UBC Theses and Dissertations

Determination of the focal points of an electronic lens by a graphical method Covington, Arthur Edwin 1940

Your browser doesn't seem to have a PDF viewer, please download the PDF to view this item.

Item Metadata

Download

Media
831-UBC_1940_A8 C7 D3.pdf [ 5.69MB ]
Metadata
JSON: 831-1.0103732.json
JSON-LD: 831-1.0103732-ld.json
RDF/XML (Pretty): 831-1.0103732-rdf.xml
RDF/JSON: 831-1.0103732-rdf.json
Turtle: 831-1.0103732-turtle.txt
N-Triples: 831-1.0103732-rdf-ntriples.txt
Original Record: 831-1.0103732-source.json
Full Text
831-1.0103732-fulltext.txt
Citation
831-1.0103732.ris

Full Text

DETERMINATION OF THE FOCAL. POINTS OF AN ELECTRONIC LENS BY A GRAPHICAL METHOD. by ARTHUR EDWIN COVINGTON. A Thesis submitted i n P a r t i a l F u l f i l m e n t of The Requirements f o r the Degree of MASTER OF ARTS. :In the Department of THE BNITERSITT OF BRITISH COLOMBIA. A p r i l 1940. TABLE OF CONTENTS Chapter I , Page, Introduction. (a). General Discussion and Purpose. 1, Chapter I I . Theory. (a) .Cardinal Points of a Lens. (b) . Methods of Ray Tracing. (c) . Few Method of Ray Tracing. 3. 7. 13, Chapter I I I . Experimental and Tests. (a) . P l o t of F i e l d D i s t r i b u t i o n w i t h a i d of E l e c t r o l y t i c Tank. 18. (b) . A P a r t i c u l a r Traoe. 24. (c) . Experimental Tests. "' " 31. 1. • DETERMINATION OF THE FOCAL POINTS Off AN ELECTRONIC LENS BY A GRAPHICAL METHOD. . . .. Chapter I . Intr o d u c t i o n In o p t i c a l p r a c t i c e methods of ray t r a c i n g have long been used to determine the o p t i c a l p r o p e r t i e s of the glass l e n s . During the l a s t few years, s e v e r a l methods have been developed by d i f f e r e n t workers f o r t r a c i n g the paths of electrons through the lenses formed by the e l e c t r o -s t a t i c or electromagnetic f i e l d s . I n t h i s t h e s i s only the e l e c t r o s t a t i c lens possessing r a d i a l symmetry w i l l be discussed; and i n p a r t i c u l a r the lens formed by two adjacent c o a x i a l c y l i n d e r s of equal diameter* In ray t r a c i n g , a l l methods whether a n a l y t i c a l or g r a p h i c a l are based upon an experimental determination of the p o t e n t i a l v a r i a t i o n along the o p t i c a x i s , except i n the case of two c o a x i a l c y l i n d e r s of the same diameter w i t h n e g l i g i b l e gap separation. This s p e c i a l case has been . ' 1 solved mathematically. Experimentally, the d i s t r i b u t i o n of p o t e n t i a l i s obtained by p l o t t i n g the l i n e s of eq u i -p o t e n t i a l formed by a l a r g e scale model of the electrode system immersed i n an e l e c t r o l y t i c tank. Since the f i e l d produced by the electrode system possesses r a d i a l (1). F. Gray, B e l l System Tech. Jour. V o l . 18, 1, 1939. symmetry , one need only p l o t the l i n e s on the surface of the e l e c t r o l y t e provided the a x i s of r o t a t i o n l i e s upon the surface. The methods of ray t r a c i n g so f a r developed are based on an electrodynamical treatment w i t h a n a l y t i c a l or g r a p h i c a l s o l u t i o n and i n one case the problem i s solved from a p u r e l y o p t i c a l point of view. The purpose of t h i s t h e s i s i s to c a l c u l a t e from a knowledge of the e q u i p o t e n t i a l p l o t the f o c a l points of the e l e c t r o - s t a t i c lens and to check these against the experiment-a l l y found f o c a l distances and p r i n c i p a l p o i n t s . The method-used c o n s i s t s of a g r a p h i c a l s o l u t i o n of the d i f f e r e n t i a l equation of the el e c t r o n ' s t r a j e c t o r y . Chapter I I . Theory (a); Cardinal,Points, of a Lens. In an i d e a l lens system, planes i n the object space are reproduced i n the .'image space as planes; s t r a i g h t l i n e s as : s t r a i g h t l i n e s M a t h e m a t i c a l l y such a transformation i s c o l l i n e a r . The constanfeof the r e l a t i o n s h i p determine the s i x c a r d i n a l points of the "lens;. "namely,-' the two f o c a l p o i n t s , the two p r i n c i p a l p o i n t s and the two nodal p o i n t s . .The'well known r e -l a t i o n s between the var i o u s q u a n t i t i e s such as object distance, image d i s t a n c e , f o c a l lengths, etc., w i l l be discussed f o r an i d e a l lens system and l a t e r , s l i g h t l y modified to include the e l e c t r o n i c l e n s . I X \ 1/ / RAY I. Figure (1). In f i g u r e (1), the p r i n c i p a l p o i n t s of a lens with the geometrical center placed at M. are shown and two rays are drawn. These rays are drawn from geometrical considerations and completely define the two f o c a l p o i n t s and the two p r i n c i p a l p o i n t s . Ray ( l ) leaves the object p a r a l l e l to the a x i s and upon entering the lens boundary at B w i l l have i t s path a l t e r e d . I f the index of R e f r a c t i o n of the lens changes continuously as i n the case of the e l e c t r o n i c l e n s , the d i r e c t i o n of the ray w i l l change continuously u n t i l i t reaches the other boundary a t B', where i t continues to move i n a s t r a i g h t l i n e i n the same d i r e c t i o n as the tangent to i t s path a t the boundary and e v e n t u a l l y i n t e r s e c t s the o p t i c a l a x i s a t F'* The i n t e r s e c t i o n of the o r i g i n a l path produced forward w i t h the s t r a i g h t p o r t i o n of the changed path produced backwards 'defines the f i r s t p r i n c i p a l plane. The a c t u a l path o f the ray through tblae lens i s shown i n dotted l i n e s . Ray (2) i s constructed i n a s i m i l a r manner exoept that i t enters the lens from the image side. In the d i s -cussion, the undashed l e t t e r s r e f e r to p o i n t s and distances i n the object space, while the dashed l e t t e r s r e f e r (to the corresponding q u a n t i t i e s i n the image space. From considering s i m i l a r t r i a n g l e s , one obtains y x - f ' from which Newton's formula i s derived: x x 1 f f' Now l e t s = f + x and s' = f ' + x' denote the object and image distance from the p r i n c i p a l points,, then Newton's Formula becomes ( s - f ) ( s' ~ f ' ) = f f ' (2) 5. The angular m a g n i f i c a t i o n "^an ^ and the l a t e r a l magni-i ' tan u f i c a t i o n £ are connected by Lagrange's Law ; n • tan 6- y * n' • tan 6 • y' where n and n' are the index of r e f r a o t i o n of the object and image space r e s p e c t i v e l y . One also obtains from t h i s law The path of an e l e c t r o n through a p o t e n t i a l f i e l d can be i d e n t i f i e d with the path of a ray of l i g h t provided the index of r e f r a c t i o n be chosen so that n - k J v where v i s the p o t e n t i a l ; and consequently, the equations worked out i n o p t i c a l theory may now be a p p l i e d to the e l e c t r o n i c l e n s . The r a t i o _JL_ where v i s the voltage i n the object space and v 1 the voltage i n the image space i s termed the voltage r a t i o and i s an important r a t i o i n the d i s c u s s i o n of any e l e c t r o n i c l e n s . The r a t i o n/n! may now be replaoed by the r a t i o v/v'. Again from s i m i l a r t r i a n g l e s one obtains s* _ f ' and s _ f y + y' y y + y ! ~. ~yT and therefore s 1 _ f' y' _ f v 1 ' mag^ (5) s. " f y " V v S u b s t i t u t i n g i n equation (3) mag.+ 1 = s 1 _ MI'. + MP' (6) f« f where "mag." i s the m a g n i f i c a t i o n produoed by the lens and i s the r a t i o of image s i z e to object s i z e ; MI' i s the distance from midpoint M to the image I 1 ; MP' i s the distance from the midpoint to the second p r i n c i p a l plane and f ' i s the 6. second f o c a l ..length. Equation (6) just derived w i l l be used to determine the f o c a l length of an e l e c t r o n i c l e n s . For a given voltage r a t i o a p p l i e d to a given l e n s , the distances MP' and f' w i l l remain constant; and i f the- object distance i s v a r i e d , then the image w i l l vary i n s i z e and p o s i t i o n and i t w i l l be noticed that the v a r i a t i o n between the m a g n i f i c a t i o n and the image distance i s l i n e a r . The two q u a n t i t i e s "mag" and MI' are measured for various p o s i t i o n s of the object and p l o t t e d on graph paper* These p o i n t s should l i e on a s t r a i g h t l i n e since equation (6) may be w r i t t e n ; MI' = f 1 mag. + f ' - MP'. The slope of t h i s l i n e w i l l give f' the second f o c a l l e n g t h , and the i n t e r c e p t on the y a x i s w i l l give f'-MP 1 and hence MP' can be found since f i s known. S i m i l a r l y by combining equations (3) and (5) one obtains an equation i n v o l v i n g the f i r s t f o c a l l e n g t h . ( 1 + - ^ a i r ) * • - « = PM + PI (7). This method of f i n d i n g the f o c a l p o i n t and the 1 p r i n c i p a l p o i n t was used by P e t r i e and i s an extension of 2 the method used by E p s t e i n . I t w i l l be noticed t h a t the equations developed apply only to the p a r a x i a l rays. (1) . D. P. R. P e t r i e , Report of Research f o r Year Ending May 31st, 1936, T r i n i t y College, Cambridge. (2) . D. W. E p s t e i n , Proc. I n s t . Rad. Eng. 24, 1095, 1936. 7. (b) Methods of Ray Tracing. The methods of ray t r a c i n g that have been developed can be d i v i d e d i n t o two c l a s s e s ; those based upon eleotro -dynamical theory and those based upon an equivalent o p t i c a l o p t i c a l system and t r e a t e d from; an o p t i c a l viewpoint. 1 Most of the o r i g i n a l work was done by J . P i c h t and 2-by Scherzer and Bruche • •- and p a r t of t h i s work w i l l be included to form a working bas i s f o r the d i s c u s s i o n on the c o - a x i a l c y l i n d e r s . Consider an e l e c t r o n , ( mass m and charge e ) moving w i t h v e l o c i t y v.*. along the z a x i s . Then the energy acquired can be w r i t t e n -§• m v =• e Q>{z). The p o t e n t i a l f i e l d i s w r i t t e n as a f u n c t i o n of z and r , namely d> (z,r) where z i s measured along the o p t i c a x i s and r i s the r a d i a l distance from the a x i s . Along the axis the p o t e n t i a l v a r i a t i o n i s (z,0) - j>{z) where (z) i s measured from the point where the electrons have zero K i n e t i c Energy. The r a d i a l force equation may be w r i t t e n ; d r _ _ e d 11 . • m yy-d z d / d z d r \ e "dT^~ ~d~z \^  d t d~*z)- m E ^ y y 3 z i m ^ The number of l i n e s of force which leave the c y l i n d e r must be equal to 4TTcV> where ^ i s the charge i n s i d e the c y l i n d e r ( i ) J . P i c h t , Ann. Physik, 15^ 926, 1932. (2) E. Bruche and 0. Scherzer, Geometrisohe Elektronenoptik, B e r l i n , 1934. 8. measured i n E. M. U. Therefore, 2 T r 1 E A+ r"TT = r V 1 4 T T c ^ from which a value of B ^ i s obtained, namely; \~ z r^4 t t o ^ -^zT^j Now E = - l i i ^ l and v, = \z~®. o{z) and the r a d i a l force equation becomes - r / T T r % r (8) az/ •"•^ ,"/" " •' * ^ This equation when solved gives the path of the p a r a x i a l electrons and when t h i s i s known, the p r i n c i p a l p o i n t s of the e l e c t r o n i c lens may be found. (a) The equation i s u s u a l l y solved by successive approximations g i v i n g r as a converging power s e r i e s . I n t e g r a t i o n y i e l d s f dT v_ dz , J ^ r ' - j & r j • — u The s u f f i x "a" denotes values of the v a r i a b l e s assumed i n the object space at z^ where the value of ^ i s n e g l i g i b l e , i . e . at the boundary of the lens where the e l e c t r o n has not yet entered the e l e c t r i c f i e l d . Another i n t e g r a t i o n y i e l d s F i r s t a value f o r r i s s u b s t i t u t e d i n the second i n t e g r a l on the r i g h t s i de of the equation which more or l e s s corresponds to the path; t h i s u s u a l l y amounts to s e t t i n g r equal to r . The equation can now be solved f o r r and t h i s value i s used i n the next approximation. The process i s repeated and the convergence of the s e r i e s depends on how w e l l the o r i g i n a l f u n c t i o n was chosen. P r dz dz 9. The s e r i e s -obtained i s r =• r, r. + Jlj> r 1 1 5* P r,, dz dz From geometrical considerations, one can ob t a i n a simple expression f o r the f o c a l length; 1 _ _ f r ~ where c< i s the angle of d e v i a t i o n and r 0 i s the r a d i a l distance before entering the lens and xx i s the value of the slope of the path on the e x i t s i d e . (b) P e t r i e used the second order approximation to determine the f o c a l l e n g t h of two c o - a x i a l c y l i n d e r s and evaluated the i n t e g r a l s that appeared by mechanical means. r 1 _ 1 - / ±- I d z d z d z (9). Where i s the p o t e n t i a l i n the image space and z a and r£_' - x- — — — — o ~ — "a are the boundaries of the e l e c t r i c f i e l d . Expressions f o r the f i r s t f o c a l l e n g t h and the p r i n c i p a l points can be developed, but w i l l not be given since they are rather com-p l i c a t e d and further, more no use w i l l be made of them i n t h i s t h e s i s . ( c ) . The f i r s t order approximation appears f r e q u e n t l y i n the l i t e r a t u r e and can only be s a t i s f a c t o r i l y a p p l i e d to a 10. t M n lens and gives f o r the second f o c a l length (6b •*>• 1 = l 2- 4 z (10), Since the space on each side of the le n s i s f i e l d f r e e , t h i s 1 equation was int e g r a t e d p a r t i a l l y by Scherzer y i e l d i n g a more workable equation. The density of the charge i n the beam was neglected and thus from equation (8), P may be w r i t t e n as - \ <j) (z) . J-G-•z. ( l i ) , Therefore 1 1__ h This f i r s t order approximation has a l s o been developed from 2 pur e l y o p t i c a l considerations by 0. Klemperer by d i v i d i n g the e l e c t r i c f i e l d i n t o a s e r i e s of t h i n l e n s e s . 3 ( & ) . W . W, Hansen and D, L. Webster have derived the f o o a l lengths of e l e c t r o - s t a t i c lenses by means of a power s e r i e s computed i n powers of the quantity Y w h i c h i s defined as the r a t i o of the P. D. across the lens to the sum of the P. D. between the ele c t r o d e s ; the square of t h i s r a t i o being neglected i n comparison w i t h u n i t y . (1) . H. Johannson and 0. Scherzer, Z e i t s . f . Physik. 80, 183, 19 (2) . 0. Klemperer, E l e c t r o n Optics, Cambridge, 1938. (3) . V/. W, Hansen and D. L. Webster, Rev. Sc. I n s t , , 7, 17, 1936 11. / ^ 1 leju Maloff and E p s t e i n have also derived a method of approximation f o r c a l c u l a t i n g the t r a j e c t o r i e s of an e l e c t r o n between c l o s e l y spaoed e q u i p o t e n t i a l s using the foroe equations: 2 / m dT-~-as 6 s i n ^ m = | i e cos£ dt*- ds where & i s the angle between the d i r e c t i o n of the f i e l d and a x i s , r i s the r a d i a l height above the a x i s , R the average radius of curvature of both e q u i p o t e n t i a l s and s i n $ = r/R , I n t e g r a t i o n y i e l d s the equations used i n c a l c u l a t i n g the path: c)r •=.change i n r a d i a l height ~ % C, t -f- v t (12a), o>^ . - change i n a x i a l p o s i t i o n - £ G^t"1"-t- v^ t (12b). where G,= 1 £ , 1 &~ ^ and v and v are ' m ds R *- m ds R -^o the i n i t i a l r a d i a l and l o n g i t u d i n a l v e l o c i t i e s . , , 2 I f ) . 0. Klemperer and W. D, Wright have made use of o p t i c a l ray t r a c i n g i n f i n d i n g the t r a j e c t o r y of an e l e c t r o n . The e l e c t r i c f i e l d i s broken i n t o a s e r i e s of t h i n lenses w i t h the same radius of curvature as the e q u i p o t e n t i a l s and w i t h an index of r e f r a c t i o n equal to Jr' where v i s the p o t e n t i a l of the p a r t i c u l a r l e n s . Ordinary t r i g o n o m e t r i c a l ray t r a c i n g formulae are used such as developed by A. E. Conrady , (1) . I . G. M a l o f f and D.W. Ep s t e i n , Proc. I n s t . Rad. Eng. .Vol.- 22, 1386, 1954, (2) . 0.Klemperer & W.D. Wright, Proc, Phys. Soc., 51, 296, 1939. (3) . A. E. Gonrady, A p p l i e d Optics and Op t i o a l Design* Oxford, 1929. 12. 1 (g'i). L. Jacobs uses a g r a p h i c a l method based upon S n e l l ' s Law of R e f r a c t i o n at the e q u i p o t e n t i a l surfaces. In Theory t h i s i s the same method as that of 0. Klemperer. A g r a p h i c a l process to f i n d the curved path between equipotent* i a l s i s o u t l i n e d , but not used since i t i s not unreasonable to assume that the path from e q u i p o t e n t i a l surface to equi-p o t e n t i a l surface i s a s t r a i g h t l i n e when a l a r g e number of surfaces are used i n the t r a c e . , . . ' • 2 111). The ray t r a c i n g machines of Gabor and 3 Langmuir a c t u a l l y give the path of any e l e c t r o n moving through any e l e c t r i c f i e l d and are not r e s t r i c t e d to the p a r a x i a l zone as are a l l of the methods described w i t h the exception of the approximation by M a l o f f and E p s t e i n . However these machines are d i f f i c u l t to construct and operate. (1) . L„ Jacobs, P h i l . Mag. 26, 57Q„ 1938. (2) . D. Gabor, Nature, 139, 373, 1937. (3) . D. B. Langmuir, Nature, 139, 1067, 1937. 13. ( c ) . New^Method of Ray Tracing. The method of ray t r a c i n g used i n t h i s t h e s i s i s based upon a g r a p h i c a l s o l u t i o n of the fundamental d i f f e r e n t i a l equation derived i n the previous s e c t i o n , namely; where P i s now taken t i j — s i n c e the density^>is small„ This equation may be w r i t t e n i n the f o l l o w i n g form jPzS ^ - ° <«>• Which i s a second order d i f f e r e n t i a l equation w i t h c o e f f i c i e n t s r e l a t e d to the p o t e n t i a l v a r i a t i o n along the a x i s and i t s f i r s t and second d e r i v a t i v e s w i t h respect to z. The g r a p h i c a l s o l u t i o n of t h i s equation or of the more general form ( ^ U j f f ^ ^ ( x ) | ^ ^ f 6 ( x ) y -- f i x ) (14), i s discussed by V. A. B a i l e y and J . M. Somerville and w i l l be considered i n t h i s t h e s i s f o r the sake of completeness. By applying t h i s method to equation (13), one even t u a l l y obtains a curve representing the path of any chosen p a r a x i a l e l e c t r o n through the given P o t e n t i a l d i s t r i b u t i o n . I t w i l l be no t i c e d that t h i s method i s a p p l i c a b l e to any a r b i t r a r y f i e l d d i s t r i b u t i o n . (1). V.A.Bailey and J.M. So m e r v i l l e , P h i l . Mag., 26, 1, 1938. 14 Graphical S o l u t i o n of Second Degree D i f f e r e n t i a l Eq.uation. In a d d i t i o n to the given, equation (14) the i n i t i a l conditions must be given. Then a t x - x o » l e t y* y ( and On squa re p a p e r . (oO choose axes O'P' and O'Q' and dx l e t O'Q,1 serve as an a x i s f o r both/»j and z ; and l e t O'P' serve as an a x i s f o r f and x ; x being measured i n the d i r e o t i o n O'P', ^ i n the d i r e c t i o n P'O'. The x and the j scales are the same. Let 0' be the o r i g i n of a l l co-ordinates. 9' If 3 ' a ; / 0 ' C ' 1 W Now the curves are drawn: /fy a f (x) = f (z) \ T •a; Points a . ^ x, A ' '"V. X where the v a r i a b l e z replaces the v a r i a b l e x i n the c o e f f i c i e n t s i n order to avoid confusion. The i n i t i a l points A£ and A', are marked on the B'C. On transparent paper ( (3) two axes XP and XQ,'are drawn and the l i n e p -1 are drawn to the scal e of j , The curves - y a n d ^ x = ^ are s t a r t e d from the 15. i n i t i a l p o i n t s A'0 and A 1 and are continued by constructing small segments according to the value of the tangents at the poi n t i n question; values which are determined by the geometrical process.. To get a p o i n t adjacent to A1,, on the curve - y. The transparent paper (^ ) i s placed over the graph (p(J so that the p and x axes coincide and R i s over C, On RS the po i n t s A D and A, are marked o f f immediately over the p o i n t s A*0 and A'. . The transparent paper (^ } i s moved u n t i l X i s on A w i t h the q a x i s c o i n c i d i n g with the l i n e x - xe and then, the l i n e j o i n i n g A*e and A ' i s the tangent to the c u r v e ^ ( - y at the p o i n t ^ x ^ y j ) . dy To get a point adjacent to A'^  on the curve r dx . On the transparent paper ((3) XT equal to x e i s marked o f f along the q a x i s and then ((3) i s moved u n t i l the p and x axes coincide w i t h T l y i n g over the curve ipD (z) . The i n t e r s e c t i o n of t h e a x i s w i t h the r a d i a l l i n e from X which passes through A „ i s a t the p o i n t K. From s i m i l a r t r i -angles, the r e l a t i o n OK = ya%(z) i s obtained. This length 16. i s marked o f f on a s t r i p of paper, s i m i l a r l y by taking the i n t e r s e c t i o n of the axis with the r a d i a l from X to A, •when T l i e s over the C^,(z) curve, the length (OK), which i s equal to ^ ^ I z ) i s found and marked o f f on the s t r i p of paper. These two lengths are added g i v i n g ; KO + (K0),= .lp 0 (z) y 0+ (z) { ^ ) o - S This l e n g t h S i s subtracted from G'B', that i s from f ( x ) . Henoe: f i x ) - p,) - lf> U)/g) = l ^ l z ) =* B fT f = f i x ) - S. I f f ( x ) i s equal to zero, then B'Y^ equals - S and the s o l u t i o n i s s i m p l i f i e d . The transparent paper ((5) i s moved u n t i l T i s over the ^ x ( z ) curve w i t h the x and p axes coincident and on (<X) the p o i n t U" is. j marked so that O'TT^B'Y'** -S . Then i n the t r i a n g l e HXO' , the tangent of the angle UXO1 i s equal to d/dx ^dy/dx^ . The paper 13) i s moved u n t i l the q a x i s i s coincident w i t h the l i n e x= x o and X i s over A' , and then the l i n e UA' i s tangent to the curve y^z- H ; A curve constructed by s m a l l segments i s l i a b l e to both random and systematic e r r o r s . The random errors tend to balance out when a l a r g e number of steps i s taken while the systematic errors due to t a k i n g f i n i t e segments of tangents can be reduced by i n c r e a s i n g the number of steps used i n the co n s t r u c t i o n of a curve. 17 The aocuracy can he g r e a t l y increased by using the Theorem of Mean Value to construct the curve by means of Chords. This i s done by choosing the i n t e r v a l c> and "by means of the method of tangents to extend the curve from the point P o (xo,y0) t o t h e p o i n t P'0 (x^£ y' ). The tangent to the curve i s determined as usual at the poin t P£ and the chord from P 0 to P ,. (Xj+^, y, ) i s drawn w i t h t h i s slope. The procedure involves twice as many operations as that of the method of tangents f o r a given i n t e r v a l & , but t h i s i s compensated by the increased accuraoy. The e r r o r involved i n any s o l u t i o n found by t h i s method cannot be found since the s o l u t i o n to the D i f f e r e n t i a l Equation i s not known. However , t h i s e r r o r can be made of the order of one or two percent i f one i s allowed to general-i z e from the r e s u l t s of s o l v i n g a known Legendre equation: the curve constructed agreed to w i t h i n two percent of c a l c u l a t e d values. In s p i t e of the apparent complexity of the method, i t was found that the time taken to complete one step by the method of tangents was s l i g h t l y under three minutes. 1-5 V _ 4 80./V-Figure (2). 18. Chapter I I I , . Experimental and Tests. (a). P l o t of F i e l d D i s t r i b u t i o n with a i d of E l e c t r o l y t i c Tank. The determination of e l e c t r o n t r a j e c t o r i e s i n most arrangements must begin with an experimental i n v e s t i g a t i o n of the p o t e n t i a l d i s t r i b u t i o n . The exceptions to t h i s r u l e are those electrode combinations of simple geometrical form whose p o t e n t i a l d i s t r i b u t i o n s may be c a l c u l a t e d with consider-able accuracy. The diaphragm and the c o - a x i a l c y l i n d e r s with small gap separation belong to t h i s c l a s s . The determination of the p o t e n t i a l d i s t r i b u t i o n of any electrode arrangement i s done with the a i d of the e l e c t r o l y t i c tank. A l a r g e scale model of the electrodes i s constructed and submerged i n the e l e c t r o l y t e with the ax i s of r o t a t i o n l y i n g on the surface. The v»rater surface thus acts l i k e a m i r r o r and the top h a l f of the model need not be constructed. The two electrodes are connected to a bridge arrangement shown i n f i g u r e (2) ; while the f i e l d i s explored by means of a probe connected to a pantograph. This bridge i s s u p p l i e d with an a l t e r n a t i n g voltage of 1.5 v o l t s at a frequency of 480 cycles per second generated by a tuning f o r k maintained by a vacuum tube. The probe i s connected to a two stage r e s i s t a n c e coupled a m p l i f i e r and a balance i s i n d i c a t e d e i t h e r by usi n g a p a i r of telephones or a m i l l i -ammeter. I t was found important to s h i e l d the leads to the probe and to have the s h i e l d i n g grounded. 19. _The value of any e q u i p o t e n t i a l l i n e i s expressed as a percentage of the t o t a l voltage a p p l i e d across the two c y l i n d e r s . The c y l i n d e r A can he regarded as at zero voltage and c y l i n d e r B as at 100 v o l t s . The e q u i p o t e n t i a l traced out by the probe w i l l depend upon the r a t i o of the res i s t a n c e s C and D. I f C/D equals l/9,then the probe w i l l trace out the 10 percent e q u i p o t e n t i a l . The two h a l f c y l i n d e r s were constructed and mounted i n the trough and t r a c i n g s of the f i e l d were taken with a gap separation p r o p o r t i o n a l to the one used i n a c t u a l p r a c t i c e . For t h i s gap separation, the p o t e n t i a l v a r i a t i o n along the a x i s i s p l o t t e d against the distance from the midplane. Eor convenience i n applying these r e s u l t s to any s i z e d tube a l l measurements are ev e n t u a l l y expressed i n terms of a u n i t chosen as the tube radius which w i l l be designated by the l e t t e r R. The r e l i a b i l i t y of the p l o t i s dependent upon the accuracy of the c o n s t r u c t i o n of the models and upon the r e l a t i v e s i z e of the tank. The l a s t c o n s i d e r a t i o n determines the i n f l u e n c e of the r e f l e c t i o n s from the w a l l s and bottom of the tank and should, be reduced to a minimum by having the model away from the sides of the tank. The accuracy can be checked from a study of the f i e l d based upon any e x i s t i n g Figure ( 3). ••.•v.--0O77J777 1 PERCf tet R ilO-BDIE TIO or p tiBALiOE OTEHTIAL ALONG A (IS7 907777J7 90^7-^ • 70-—— 60-—— _{-'}:'-: 50 40 •30 -Z0 --10-P15T net FR )ii -MI; PlAHt r " 5 - 0 -c 5 0 C 5 HO 1 5 2 L Figure (3). 20. symmetry. • As an example; the 50 percent e q u i p o t e n t i a l l i e s between the two c y l i n d e r s and i s a plane and hence i n the tracings should appear as a s t r a i g h t l i n e . However, i t was found that i f the probe were connected d i r e c t l y to the g r i d of the f i r s t a m p l i f i e r tube, then t h i s e q u i p o t e n t i a l was n o t i c e a b l y curved. This displacement was a t t r i b u t e d to a small voltage produced by the' g r i d current f l o w i n g through the e l e c t r o l y t e and consequently throwing the bridge out of balance. This trouble was remedied by connecting the probe to the a m p l i f i e r by means of a transformer which had the center tap grounded. I t was found that the nature of the free ends of the c y l i n d e r s , whether l e f t open or closed, d i d not produce any noticeable e f f e c t upon the p o s i t i o n s of the e q u i p o t e n t i a l s c o n s t i t u t i n g the main p a r t of the lens provided the tubes were a t l e a s t 2 R u n i t s i n length. The e l e c t r i c f i e l d which thus c o n s t i t u t e s the lens i s 4 R u n i t s long and no object can be brought i n s i d e t h i s distance without s e r i o u s l y d i s t u r b i n g the p o t e n t i a l d i s t r i b u t i o n and hence the p r o p e r t i e s of the l e n s . 21. Results. ' > The p o t e n t i a l v a r i a t i o n along the a x i s of two adjacerft c o a x i a l c y l i n d e r s w i t h n e g l i g i b l e gap separation and with a p p l i e d voltages v, and vx has been developed by Gray and i s given by the equation; • equation to be appl i e d to the c y l i n d e r s when the gap separation i s f i n i t e . This was done i n the f o l l o w i n g manner. The model cy l i n d e r s w i t h a radius of 7 centimeters were set i n the tank w i t h a gap separation of 0.84 centimeters: the r a t i o of gap separation to radius,0.12 chosen to correspond w i t h that used i n the a c t u a l l e n s . The r a t i o of the r e s i s t a n c e s C and D were now set so that the probe would trace out an e q u i p o t e n t i a l of given value. I n t h i s way by v a r y i n g the r a t i o C/D the whole f i e l d was explored and gave the p o s i t i o n s of the various e q u i p o t e n t i a l s as shown i n f i g u r e ( 5 ) . From symmetry, the absolute value of the a x i a l i n t e r c e p t of the 50 percent and of the 70 percent e q u i p o t e n t i a l should be the same. Both i n t e r c e p t s were measured from the midplane and averaged; thus y i e l d i n g a value of z ! which was s u b s t i t u t e d i n t o equation (15) and then solved f o r the constant . A graph constructed from data shown i n Table I shows the a x i a l p o t e n t i a l v a r i a t i o n against midplane distances. This graph i s shown i n f i g u r e (4). Only the experimentally found p o i n t s are entered i n order to avoid oonfusion with c a l c u l a t e d values of the p o t e n t i a l where LO - 1.32~(15) Readjustment of the constant to permitted the 22. since there i s good agreement between the two sets of values, although the constant d i f f e r e n c e s which can he seen from i n s p e c t i o n of the l a s t column of Table I show that the errors are not a c c i d e n t a l but a r i s e from approximating the a x i a l p o t e n t i a l v a r i a t i o n by a modified form of equation (15). The greatest departure, about 0.4 percent, occurs at the end of the curve where the p o t e n t i a l i s close to that of the adjacent c y l i n d e r and i s n e g l i g b l e since an error of t h i s magnitude can e a s i l y be introduced i n drawing a smooth curve between given p o i n t s . Moreover, the o v e r a l l accuracy of the e l e c t r o l y t i c tank and the associated apparatus w i l l be of t h i s order. For the purpose of working, equation (15) w i l l be put i n t o a more s u i t a b l e form by having v(=- 0, namely; , <£i|! JL= = i ^ l - v tanhu> z') L, Jacobs ^ has also used Legendre polynomials to o b t a i n an approximation to. w i t h i n one percent of any a r b i t r a r y p o t e n t i a l v a r i a t i o n along the a x i s . U ) . L. Jacobs, P h i l . Mag, V o l . 26, 601, 1938. Table I . 23 c EXPERT ME MTAL. 50$ 45$ 40% 30$ 25% 20 'JO 15$ 10$ 2%r AXIAL ITHTERCE-PT -0 o60 -1.10 -1.70 -2.35 -3.00 -3.80 -4.85 -6.30 -8.50 -10.8 55$ 60% 65$ 70$ .75$ 80$ 85$ 90$ 95$ 97£ AX 1 A L . I N T E R C E P T 0 0.55 1.10 1.75 2.35 3,05 3.85 4.85 6.20 8.50 10.8 A V E R A G E 0.575 1.10 1.725 ,35 3.025 3.825 4.85 6.25 8.50 10.8 V o 0^082 0.157 0.246 0.336 0.432 0.546 0.694 0.894 1.214 1.543 Jo 0 0.103 0.196 0.308 0.420 0.540 0.683 0.868 1.118 1.518 1.929 TArttt to^' 0 .1027 .1938 .2984 .3969 .4930 .5935 .6905 .8067 .9083 .9586 CALCULATED 50.0$ 55.1$ 44*9$ 59.7$ 40.3$ 64.9$ 35.1$ 69.9$ 30.1$ 74.7$ 25.4$ 79. 7$ 20.4$ 84.6$ 15.4$ 90.3$ 9.7$ 95.4$ 4. 6$ 97.9$ 2.1$ Columns 1 and 2 l i s t the percentage voltage of the eq u i p o t e n t i a l s . 24. (b) A P a r t i c u l a r Trace. The p r i n c i p a l points and the f o c a l points of an e l e c t r o n i c lens formed by two c y l i n d e r s with a gap separation of. .12 R are found by t r a c i n g two rays through the e l e c t r i c f i e l d using the method o u t l i n e d i n a previous s e c t i o n . been found experimentally from a use of the e l e c t r o l y t i c tank and i t was found that i t could be approximated by an a n a l y t i c f u n c t i o n <ji (z')» Since the f i r s t and second d e r i v a t i v e s are needed, t h i s f u n c t i o n can be d i f f e r e n t i a t e d by u s i n g ordinary c a l c u l u s methods. However, i f the p o t e n t i a l f u n c t i o n cannot be represented by some a n a l y t i c f u n c t i o n , considerable d i f f i c u l t y may be experienced i n f i n d i n g the two d e r i v a t i v e s . Recourse could be made to g r a p h i c a l methods once the p o t e n t i a l v a r i a t i o n along the a x i s has been p l o t t e d but a b e t t e r method would be to make use of the Calculus of F i n i t e D i f f e r e n c e s . The correctness of the d e r i v a t i v e s can be checked by forming the r a t i o 2. which should equal the radius o f curvature of the e q u i p o t e n t i a l at the p o i n t i n question on the a x i s and t h i s can be measured by using c a l i b r a t e d templates . I t should, a l s o be noted that 2. ^ " e q u a l s some d i f f i c u l t y and must be watched. In the d e r i v a t i o n of However, i f the e l e c t r o n enters the f i e l d w i t h a v e l o c i t y The p o t e n t i a l v a r i a t i o n along the o p t i c axis has the fundamental was made zero when v, - 0 . 25 . gained, i n f a l l i n g through a p o t e n t i a l of <j>^ then (^yj -must he used i n the equation. In the case of the c o a x i a l c y l i n d e r s <j> (z) -a-f-b t a n h i o z 1 where z' i s measured from the midplane: a - — & h = •Y' „ When z' i s equal to zero, then <j> ts') i s equal to the p o t e n t i a l of the midplane and obviously a change of v a r i a b l e s i s necessary. I f z T i s put equal to -2.5 R, then the p o t e n t i a l (j> (z') i s equal to the p o t e n t i a l o f the f i r s t c y l i n d e r f o r most p r a c t i c a l purposes and corresponds to . Hence a proper o r i g i n i s obtained by i n t r o -ducing the new v a r i a b l e , z - z ' f2.5 or z ! = z - 2.5 and the correct p o t e n t i a l equation i s <^(z) - a +- b tanh u>{ z -2,5). The absolute value of the p o t e n t i a l does not change the o p t i c a l p r o p e r t i e s of a lens ( disregarding space charge e f f e c t s ) since i f the p o t e n t i a l i s m u l t i p l i e d by a constant d i f f e r e n t from zero, the d i f f e r e n t i a l equation remains unaltered. Hence any, convenient scale f o r the p o t e n t i a l can be taken. S i m i l a r l y , i f " r " i s m u l t i p l i e d by a constant, the roots of the s o l u t i o n of the equation remain unalt e r e d and hence the f o c a l p r o p e r t i e s of the l e n s . I t i s not to be thought that the f o c a l l e n g t h i s the same f o r a l l rays entering the system at various r a d i a l d istances, that i s to assume that there i s no s p h e r i c a l a b e r r a t i o n f o r i t must be remembered that the d i f f e r e n t i a l equation i s r e s t r i c t e d to p a r a x i a l rays. However f o r ease i n c a l c u l a t i o n , any convenient r a d i a l distance i s taken and c a l l e d u n i t y . I t has been shown before that the o p t i c a l p r o p e r t i e s of a lens are completely determined by two rays entering the the system, from opposite sides and v^ith t h e i r i n i t i a l d i r e c t i o n s p a r a l l e l to the o p t i c a l a x i s . Two such rays are traced and hence the i n i t i a l conditions f o r a s o l u t i o n of the d i f f e r e n t i a l equations are: ray (1); z ~ 0 r = r 0 = 1 r J -=• 0 ray (2); z = 5 r - r e = 1 r£ = 0 The c u r v e s , Jt and^p0 were p l o t t e d on a sheet of inch graph paper, twenty inches square, where - § (z)• » a + h tanh <o ( z - 2.5 ) ft = -|- *-| - i co s e c h ^ ( z - 2.5 ) f .=• £ J l f , = - i to"sech^(z-2.5) tanh to (z-2.5) Since the absolute p o t e n t i a l i s immaterial, the d i f f e r e n c e Vx - v, = h was taken to "be equal to u n i t y . A voltage r a t i o 2 v,/ Y%equal to l / 5 was used and therefore XT, & - Y^ + V/ _ i _ i _ f ^ - 3 b v^ _ - v, 1 _ A T . " 2 / i s . . g i v i n g a - 5/2 b = 1 . 5 . The curves Jt and^, were constructed from data shown i n Table I I . Two curves are constructed by the method o u t l i n e d i n Chapter IT f o r a s i n g l e e l e o t r o n entering the lens p a r a l l e l to the a x i s ; one showing the r a d i a l distance of the e l e c t r o n against the a x i a l distance, i . e . the a c t u a l t r a j e c t o r y ; and the other showing the d e r i v a t i v e of r wi t h respect to z. Ray (1) i s the path of an e l e c t r o n entering the lens from the low p o t e n t i a l s i d e , while ray (2) i s the path of an e l e c t r o n entering the lens from the high p o t e n t i a l s i d e . Table I I f. 5.0 4.5 4.0 3.75 3.5 3.25 3.1 3.0 2.9 2.8 2.7 2.5 2.3 2.2 2.1 2.0 1.9 1.75 1.5 1.25 1.0 0.5 0 2.5 2.0 1.5 1.25 1.0 0. 75 0.6 G.5: 0.4 0.3 0.2 0 M3.2 -0.3 -0.4 -0.5 -0.6 -0.75 •1,0 -1.25 •1.5 2.0 2.5 3.12 2.50 1.87 1.56 1.25 0.937 0.750 0.625 0.500 0.375 0.250 0 -0.250 -0.375 -0.500 -0.625 -0.750 -0.937 -1.25 -1.56 -1.87 -2.50 -3.12 .9960 . 9866 ,9536 .9154 .8483 . 7538 .6352 .5545 .4621 .3584 .2449 0 -.2449 •.5584 .4621 -.5545 6352 .7358 •.8483 .9154 .9536 .9866 9960 2.496 2.487 2.454 2.415 2.348 2.234 2.135 2.055 1.962 1.859 1.745 1,500 1.255 1.142 1.038 0.945 0.865 0.764 0.652 0.585 0.546 0.513 0.504 .00485 .0166 .057 .101 .176 .289 .372 .434 .485 .545 .588 .625 .588 .545 .485 .454 .572 .289 .176 .101 .057 ,0166 .00485 -.006 -.0205 -.068 -.116 -.186 -.265 -.295 -.500 -.280 -.244 -.180 0 .180 .244 .280 .500 ,295 .265 .186 .116 .068 .0205 .006 Figure (5). 28. Simple c a l c u l a t i o n s from data given "by these curves shown to s c a l e i n Figure (5) enables one to o b t a i n the f i r s t and second f o c a l lengths as w e l l as the f i r s t and second p r i n c i p a l p o i n t s . Ray (1): at z =• 0 , r - 1 dr/dz - 0 at z~5 , r^,341 dr/dz = -.122 Ray (2): at z - 0 , r=.914 dr/dz - .257 • at. • z = 5 , r = 1 dr/dz - 0 C h a r a c t e r i s t i c s of the Lens. F i r s t f o c a l length 3.9 R Second f o c a l l e n g t h 8.2 R Square of the r a t i o f z / f ( 4.4 F i r s t p r i n c i p a l plane from Midplane -2.1 R Second p r i n c i p a l plane from Midplane -2.9 R F i r s t m i d f o c a l l e n g t h 5.3 R Second midfocal length 6,0 R The c o n s t r u c t i o n of the curves was c a r r i e d out by means of the method of chords: ray (1) i n 19 steps and ray (2) i n 24 steps. The i n t e r v a l s were not of equal distances f o r i t was found that long steps could be taken when the curves were changing sl o w l y and that shorter steps were necessary i n the r e g i o n of maxima or minima. For the sake of comparison and as a check, the second f o c a l l e n g t h has been c a l c u l a t e d using some of the other methods. 29. (a). The f i r s t order approximation formula can he e a s i l y found since the i n t e g r a l can be r e a d i l y evaluated. This method y i e l d s — - 1 ^ — = — w h e r e LO - 1.25 and 3 ( 1 +- y*) V l y v a. For the case of the c o a x i a l c y l i n d e r s under d i s c u s s i o n , where y -1/5 , the second f o c a l length i s found to be 11.4. (b) . The i n t e g r a l appearing i n the formula of Scherzer was evaluated mechanically from data appearing and derived from Table I I and gave a f o c a l length of 11.2 R u n i t s N a t u r a l l y t h i s i s the same value as given by the f i r s t order approximation since one i s derived from the other. (c) . The f o c a l l e n g t h can a l s o be found using the second order apDroximation given by the expression; The f i r s t i n t e g r a l has already been evaluated and now the second t r i p l e i n t e g r a l must be found. This was done by mechanical means wit h the a i d of curves constructed from data appearing i n Table I I . i * ^ L r JO.1375 + 0.05951 - 0.1245 Hence f = 8.04 R. 30. (•&). The formula of Hansen and Webster gave very-poor r e s u l t s and so w i l l not be given. This was to be expected since the square of the r a t i o i s 0.443, a quantity which i s by no means n e g l i g i b l e i n comparison to u n i t y . teen regions so that the r a t i o of the i n d i c e s of r e f r a c t i o at the various surfaces was constant equal to 1.05916. A f t the manner of Klemperer and Wright a p a r a x i a l ray with an i n i t i a l r a d i a l distance of 0.1 R was traced through t h i s system of lenses u s i n g standard ray t r a c i n g formulae. This method gave a f o c a l l e n g t h of 8.08 R. (e). The p o t e n t i a l f i e l d was subdivided i n t o f. •31. (o). Experimental Tests. In order to t e s t the preceeding theory and c a l c u l a -t i o n s , two c y l i n d e r s of 1 centimeter diameter were constructed and separated by a mica washer 0.6 m i l l i m e t e r s t h i c k . At the low voltage s i d e , a wire screen w i t h a spacing of 0.25 mm. was placed a known distance from the gap which formed the e l e c t r o n i c l e n s : and at the h i g h voltage side a fluoresoent screen was fastened to the end of a s t i f f wire which was moved from outside by an electromagnet. The wire screen was i l l u m i n a t e d by an e l e c t r o n beam formed by a simple immersion o b j e c t i v e . The electrode arrangement i s shown schematically i n f i g u r e (6). F i L A n E r i T j 33 8 J E C T , T T - , J —I \ v i i ' J / Figure (6). The immersion o b j e c t i v e consisted of three copper d i s c s and a tungsten ribbon, 0.5 m i l l i m e t e r s wide used as a f i l a m e n t . p l a c e d between the f i r s t and second diaphragms. The holes i n the second and t h i r d d i s c s are 0.7 mm. i n d i a -meter and were worked i n t o shape by punching and f i l i n g . The f i r s t d i s c , although not e s s e n t i a l i n the immersion o b j e c t i v e was very u s e f u l i n c o n t r o l l i n g the beam i n order to Figure (7). Figure (8). 32, o b t a i n more, uniform i l l u m i n a t i o n . This p l a t e , known as the backing p l a t e , i s used to s t r a i g h t e n out the e q u i p o t e n t i a l s at the surface of the filament. I f the p o t e n t i a l of t h i s p l a t e i s not c o r r e c t l y adjusted the curved e q u i p o t e n t i a l s i n f r o n t of the filament act as a c y l i n d r i c a l lens and consequentally narrow or widen the image. The f i r s t and second p l a t e s were separated 7 m i l l i -meters, the second and t h i r d p l a t e s by 5 m i l l i m e t e r s : and best r e s u l t s were found when the filament was 2 m i l l i m e t e r s from the second p l a t e . The object was mounted on a d i s c which could move i n a s l o t t e d sleeve attached to the t h i r d p l a t e and f i r s t c y l i n d e r . The r e g i o n beyond the second c y l i n d e r i s made f i e l d f r e e by a long l i g h t sleeve of brass shim, which can be attached to the second c y l i n d e r . The fl u o r e s c e n t screen moves i n s i d e t h i s sleeve and no d i r e c t e l e c t r i c a l connection i s made to the screen since i t soon aoquires a p o t e n t i a l some few v o l t s below that of the adjacent sleeve. A photograph of the i n s i d e tube i s shown i n f i g u r e (7) and a olose up photo-graph of the electrode arrangement i n f i g u r e (8). The tube containing the electrodes was mounted i n s i d e Hehmholtz c o i l s adjusted to n e u t r a l i z e the earth's magnetic f i e l d , since otherwise the beam of electrons would not stay on the o p t i c a x i s . Construction of the Tube. The e n t i r e tube was constructed from brass except f o r the i n s u l a t o r s , the l e a d - i n stem f o r the wires to the electrodes and f o r the observation windows. The outside tube 33. was made of brass tubing, 10 inohes long and 2 inches i n diameter, and w i t h two smaller brass tubes s i l v e r soldered on to i t : one f o r evacuation and the other for observing the p o s i t i o n of the f i l a m e n t . At e i t h e r end, a heavy brass p l a t e i s b o l t e d to a flange shrunk over the brass tube and a vacuum t i g h t j o i n t i s made by forming a r a i s e d r i n g on the end p l a t e to f i t a hollow groove f i l l e d w ith a l e a d washer on the flange These j o i n t s were waxed and stood up to many changes. One of the end p l a t e s had a Pyrex window cemented i n to allow obser-vations of the screen and a l s o a glass tube to permit the wire h o l d i n g the f l u o r e s c e n t screen to be moved by an e l e c t r o -magnet. The other end p l a t e had syphon bellows to receive the glass stem f o r the tungsten seals and also a l i g h t metal frame work to hold three quartz rods which support the e l e c t -rodes separated by small glass washers ground to the proper s i z e . Mounting the electrodes on three rods ensured that they were mechanically centered. The necessary vacuum was obtained by an o i l d i f f u s i o n pump backed by an ordinary Oenco r o t a r y pump. Steady voltages were obtained by u s i n g a bank of B b a t t e r i e s to supply 380 v o l t s to the f i r s t c y l i n d e r and by using a neon bulb s t a b i l i z e d power pack to supply 1900 v o l t s to the second c y l i n d e r , These voltages were measured by the current flow through f i v e 400,000 ohm r e s i s t o r s connected i n s e r i e s , each w i t h an accuracy of 1 percent. The image on the f l u o r e s c e n t screen was viewed by means of a cathetometer. 34. D i s c u s s i o n of Results. The theory underlying the method of measuring "the f o c a l lengths and the p o s i t i o n s of the p r i n c i p a l planes has been o u t l i n e d i n Chapter I I and makes use of the equations p * f^ ( m + 1 ) - MP' (16a) q - f, ( 1 + - ) - MP (16b) ^ ' m where p and q. are r e s p e c t i v e l y the image and object distances from the midplane. The l o c a t i o n of the image was extremely d i f f i c u l t since the depth of focus i s large and the usable aperture i s l i m i t e d by s p h e r i c a l a b e r r a t i o n . I t was found that only one tenth.of the aperture could be used and when the. c y l i n d e r s are 1 centimeter i n diameter the p o r t i o n of the object f a i t h f u l l y reproduced i s very small, somewhere of the order of 0.1 centimeter. One i s also bothered by a c e r t a i n amount of l i g h t from the filament which f a l l s on the screen thus decreasing the contrast between the image and the background. A l l the c a l c u l a t i o n s have been made i n a u n i t equal to one tube r a d i u s , which thus necessitates measuring a l l object and image distances i n u n i t s of 0.5 cm.. Y a r i a t i o n i n s e t t i n g the screen was of the order of 1 or 2 centimeters depending upon the p o s i t i o n of the object; and consequently no claim can be made f o r accuracy since t h i s i s an e r r o r of 4 R u n i t s . Figure (9). 35. The readings obtained are shown i n Table I I I and p l o t t e d i n Figure ( 9 ). When reduced these r e s u l t s give a second f o c a l length of 7.6 R units measured from the second p r i n c i p a l plane which i s 4.5 R u n i t s from the midplane. Table I I I . p m p m 14 R 1.5 16 H 1.64 17 R 2.0 21 R 2.2 £4 S 2.84 . 30 R 3,4 31 R 3.8 Malo f f and E p s t e i n regard t h e i r e l e c t r o n i c lens as an equivalent t h i n lens placed between the c a r d i n a l points and determine p, q and m f o r two p o s i t i o n s of the object; and then make.use of equations (16) to f i n d the unknown q u a n t i t i e s . However, the f i r s t f o c a l length i s not found d i r e c t l y from equation (16b). since the d i f f e r e n c e q^- q( which enters i n t o the s o l u t i o n f o r f, i s very small and can not be measured ac c u r a t e l y . They p r e f e r , t o derive a value fo r f ( from the measured value of f x by means of the equation f ^ jvT/ v^ f ^ . A s i m i l a r d i f f i c u l t y i n obt a i n i n g a d i r e c t value o f the f i r s t f o c a l length was encountered i n t h i s work. Another source of e r r o r which enters i n t o any work o f t h i s k i n d i s the i m p o s s i b i l i t y of d u p l i c a t i n g on a small scale the model set up i n the e l e c t r o l y t i c tank. However, f o r a simple arrangement such as the c o a x i a l c y l i n d e r s t h i s should be a minimum because of the geo-m e t r i c a l s i m p l i c i t y of the arrangement. 37. Aolaiowledgements. The v/riter wishes to express h i s thanks to ?. K. C. Mann f o r h i s advice and encouragement i r i n g the w r i t i n g of t h i s t h e s i s ; to Dr. A. M. Crook r the loan of P e t r i e ' s Report, and to Mr. W. Fraser r the cons t r u c t i o n of the experimental tube. 

Cite

Citation Scheme:

        

Citations by CSL (citeproc-js)

Usage Statistics

Share

Embed

Customize your widget with the following options, then copy and paste the code below into the HTML of your page to embed this item in your website.
                        
                            <div id="ubcOpenCollectionsWidgetDisplay">
                            <script id="ubcOpenCollectionsWidget"
                            src="{[{embed.src}]}"
                            data-item="{[{embed.item}]}"
                            data-collection="{[{embed.collection}]}"
                            data-metadata="{[{embed.showMetadata}]}"
                            data-width="{[{embed.width}]}"
                            async >
                            </script>
                            </div>
                        
                    
IIIF logo Our image viewer uses the IIIF 2.0 standard. To load this item in other compatible viewers, use this url:
http://iiif.library.ubc.ca/presentation/dsp.831.1-0103732/manifest

Comment

Related Items