Title: On the formation and collimation of an axially-symmetrical hollow electron beam
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Permanent Link: http://ufdc.ufl.edu/UF00098025/00001
 Material Information
Title: On the formation and collimation of an axially-symmetrical hollow electron beam
Physical Description: 70 leaves. : ; 28 cm.
Language: English
Creator: Adams, Leonard Caldwell, 1921-
Publication Date: 1956
Copyright Date: 1956
 Subjects
Subject: Electron beams   ( lcsh )
Electrical Engineering thesis Ph. D
Dissertations, Academic -- Electrical Engineering -- UF
Genre: bibliography   ( marcgt )
non-fiction   ( marcgt )
 Notes
Thesis: Dissertation (Ph.D.) -- University of Florida, 1956.
Bibliography: Bibliography: leaves 68-69.
Additional Physical Form: Also available on World Wide Web
General Note: Manuscript copy.
General Note: Vita.
 Record Information
Bibliographic ID: UF00098025
Volume ID: VID00001
Source Institution: University of Florida
Holding Location: University of Florida
Rights Management: All rights reserved by the source institution and holding location.
Resource Identifier: alephbibnum - 000551198
oclc - 13305459
notis - ACX5664

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Onthe Formation and Collimation
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'rl ofu~AxiPlly-Symmcaic~I Hollow Electron Beam


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ACKNOWLEDGMENT


The author wishes to express his sincere gratitude

to all members of the supervisory committee, both former

and present, for their continued guidance and encourage-

ment in this endeavor. He is especially indebted to

Dr. E. E. Muschlitz, Jr., who contributed immeasurably of

his academic and practical skills during the experimental

work and preparation of the dissertation,















a















TABLE OF CONTENTS



LIST OF TABLES ... .. .. . .. .. * *

]LIST OF ILLUSTRATIONS e . . . . .. 9

Chapter
I. INrROD~UCTION. .. . . .

I[I. THEORY RELaTING TO BEAM DESIGN . i. .; ,. 3

Condi~itions. n he4 C~athoder Reg~~ion~!.
Conditions ir -the Transitionl~ Regi a
In terpretation.: of Magn~eti Ptfta
~in Tiermsn of Flux a...
Nature ~ ~ '' of I asaFntino



Irl. t~Ap p~xroach t 'Egnettrkite 4 .94 1





e .thod .n





















TABLE OF CONTENTS (continued)


r.. Page

SAPPENDIX fII. ONJ PLUX POTTINGG . .. .. . 64

.lilBLIOGRARIPHT......... 68


,~i BIOGRAPHYY ..................... 70






*IIY







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LISaT OF TABLES


Table age .~....i:li

1. Potentials Observed in the Electrolytic Tank
Analogue .. 3

2. Tube Element; Identification ... ;. ..... 4

3. Tube Test Data ... ...;.. .

4. Collimation Data . .. .;; .I ,. -. ,....* ;: Ai'l':;i~l





LIST OF ILLUSTRATIONS


F.iigureti~q Page

S.1 Regions of the Electron Beam. ..... 4

2.2 ollow Beam Immersed in an Infinite Diode 4

j $.,3 Th Fe Magnietic Field and the Electron Beam in
tjie Transition Region ..... ...12

2,4~: :'i~ffere~ntial Quantity of Flux Passing
Through ar Differential Area 12
2 .5;;ii ;rftgi~s.. Axial Distance in the Transition


3ii~ll:i Tank SchEb~e e for Solution in the Cathode
;i egli on . . . . . . . 23
3 Typital Electrode Positions . .. .. .. 23

3I~ ;f~f i~.)TLgaskAr anemn for Takring Magnretic Field
Q~ta.................... 25
1;r~ler :j~ ,,g.;igal~j Pofte~ntial Djitribution ALong the
Ma w . . . . . . . 28

Pont out"ide thbe Beam ... 29

ia 45 E14etrodOiS~ e System For the Cathode


passeeig~ii~~i Field . .. .. .. 35
~E~iMia istnc Ein 'the Trtansitiod~
4 , .. . . . . 36

ll~l~';l;,f ;;;i' 80,1enoid C" 'rrent . . 36

o'inthi Used in.,~ SampleP Calc~ula-
;ii '' ..... 38

















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LIST Oi ILLUSTRATIONS (continuQd)
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F~gure Page
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4.6 Yop of glectric Potenti~l Field ia the

Charge Free Region i........... 11 ii iii'l
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4.7 Cross-SectiOZ1 of the AolXo* Ream Elsofroa
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4.8 The Electron Oun and Test Set-Up
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CHAPTER I

INTRODUCTION

Electron beams having high space-charge density are

of great importance in the operation of traveling-wave ampli-
fier and oscillator tubes. Those designs which have been most

successful have employed axially symmetrical beams, and a con-
siderable amount of work has been done in studying the proper-

ties of such beams.1-5*

In the main, these tubes have employed solid beams

tith as Aziq11y long interaction space (the name commonly used
to refer to the region of interaction between electron beam

and tthweding electromagnetic wave). To maintain a relatively

toDat 6 beRM radius extremely strong axial magnetic fields
Are ordinarily employed. In fact, an infinite field is fre-

ggently assumed in theoretical analyses to render difficult
problean soluble.

This work is concerned with hollow cylindrical beans
allimated by a weak radial magnetic field in conjunction with
ectr field focusing. Hartia5 has demonstrated that hollow
el otrob b ass, having a rotational as well as an azial motion,


8 pet cript homerals refer to Bibliography.








can be maintained without an axial magnetic field. Then if
such a beam were formed and properly introduced into the

interaction region, the tube would operate with a great say- .I

ing of beam and magnetic field energy.

Tangential velocities are given to the eleectfons for
two purposes. lFirst, assuming that collimation is poselble,' ;;.'
the spiral trajectory provides a lon~g interactionl space in a~;~.~~l'i!:; '
relatively short axial length. The~ slwwaesidn sr

ture can be wound about the bea~m as a hielix having the same;;;f:: ';l:3i

pi tch as the electron trajectory.; Thus ^c~ollisation& over i.' .i;I':j;:"/:!ld
great length becomes unneceissryp. Second, the cen tr furr ga; l~ll!'l91
forces are usefeul in the collimration iprobles~~ii~ particulart a~~p
the inner beam ~rdius.

In viewo of he desirble appikdations 94 p
this efforOt is a sgtudy of tedae 01a

design technique.~ for~'i an leftn a *E








Sappied. to theii














CBAPTERC II


THEORY RBLATINIG TO BEAMI DESXIG

Thei electrog beam discussed here is of hollow cylinl-
rica geomeitryi, hearing aboutt the esae size as a one-inch
wate pt~ 4P~. The~ ovner-al length of the beam is about two rad
on-hl a iPer.Fr covnveniece of analysis the beam

114:~: c Maihis de. ed to estatC in two regions.: First, the cathode re-

glen 44 pae~ e e -of abouhet one centimeter in which no magnetic

field othes. 14. this regioni the current is spacewha~rge-
M~ta8 S hid'slawis assumed to apply. Second, the

agip 'iggias fterr the beam passes through an anode
BB:a t4gtty ::of the cathoded region. In the tran-
ben paaeathrough a radial magnetic field
Ilelliei:~ty.:.l n;ii~~':i~;~~~L~''Fi~gure~ 2.1 depicts these divi-







t stgon. Ig Woalthis pro-
II~1 "'o infinite beam' ex-'
~;"i~~";sl~r ~Mamel~ field and
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boundaries of the desired portion, then the desired portion
nill remaid Collimated.

Consider Pigure 2.2. This represents a section
bettwen ~tinfinite parallel planes, one of which is a cathode

and: the other an anode wi th space-charge-limi ted current
f; low .betireen. The cylinder, shown in dark lines, repre-
;:sent the bea geometry we would like to select from the

biini~nit of charrge. Clearly, the comrponent of electric
fielsld notat.1 to the surface of the cylinder is mero. There-

to2re, it the cylinlder is isolated from the remainder of the
she~ii one coedition on its eqluilibrium is:


r 0 at r ra and r L rb, (2 )

Wh is he electr~ic potential in volts

the;;;;;;;"' "" radial co0gIouldirS.dine i meer

herP4~~~i~a:iP~S0 odtio or equilibriu of the cylinder

Ille hq lll45i~rprtial oteti distribution of the infi-
a4l~ii~Llj; thisl~ Ie mae an appliicati~ion of C hilid 's

as 83-4079 states that the current density

~B j~aiAea b I~~,um is directly piroportional to
~i~~iit~i .Wptaiand taverselyy proportion-
ae~~of sui ad-lthod sqpeta8 p. lor the
ert~~~~~~; atta of1ttin oeta


4 teepdedhere








For this case potential varies only in~the z-direc-

tion and Poisson's equation becomes


ad (2.2)
dz2 to

where p is the charge density in coulombs per cubic meter

so is the permitt~ivity of free space in faroads per
meter.
:.iI
Current continuity requires that

J - py, 2)
where J is electron current density in asppere per sque a
meter .


id

for the crae of Zero cathdh poten:QtiB-;al, sith :

eO ,



rsr-a L'9.11 x -1 i tpts h
Bolva eqet $4 do. a qpato
tu ng inii:i;~~;.; 'I ;;A ;I.i. obthii~l in';."~ ~i;di.. ;1;. I






7

Multiply this through by 240 to obtain the integrable form,
Kz

2d de 0-1/240.

Integrating,


01/2+ C1. (2 5)

For the space-charge-limited case the potential and elee-
tric field are zero at the cathode. Hence Cl is zero.
Taking the square root of both sides of equation 2.5 and

lategrating again yields

03/4- a + C2-

Eare enthode potential requires that C2 be zero, so finally


4/3 2/3 1/3
# z4/3. (2.6)


EgeAtton 2.0 states that for a dense beam having current den-
ty 4, the potential at all points along the beam is deter-


) Ez (2.7)

(ght 9,T And 2 1 84thbtled at radii a and b are the
SA And avtficient conditions that the hollow beam be
1 As $be cathode region. The electrode system must

4 4 producing these conditions with the beam pres-






8

Note that in the derivation of 2.7 the current den-

sity was treated as a constant. A notable exception to this
assumption is shot noise. This effect is due to the ten-

dency of cathodes to emit electrons in "bunches" rather than
in continuous flow. However, the space-charge has an absorb-

ing effect on these bursts of charge and minimizes current
density variations. This is one of the reasons for using

high electron densities in the tube.

Conditions in the Transition Region
The electrons are focused in the cathode section of

the gun and are accelerated to the desired Velocity by an
anode operating at an appropriate potential. The anode has
a circular opening which permits the beam to pass into the
transition region. In this section of the tube the electron

are caused to pass through a steady, radial, magnetic field,

In so doing they receive an angular component of velosity
which is very important in this tube from two standpoints.
First, it is the purpose of a gun of this type to provide a
dense beam with A long interaction path in order that anad
mum utility any be made of the tube. Beam rotation requt 6
an individual electron to travel along a spiral trajects

thus stretching tes interaction space to several fined
linear length of the beas. Second, the contritagtt t

experienced by the electreas turn out to be opit at




ili .i




In what follows the differential equations perti-
Slent to ~asltranatn region design are developed. Lagrange's
equai~tite states that


'i- '0, (2.8)
dt q

vb :~fre~ L (T U), the Lagirsangln
;T ;rRhetn tic energy of the particle

'i" U Po~E~r~tentia;LL energy of the particle such that

P!;:I li -VU- is a conservative force field
a the 1~ft coordinate of the particle
tetotal time derivative of the ith


i' rg, T, of anyr given electron is simply



.,a IB no~l;~~t; s eaLsily expresqpd. It is

I~: h potentlil such that its gradi-
e absnowof agnti flush

r; i i '...(2 .9)i

As tobecosdeedeneatv
forc 11 since






10

E represents the conventional electric field. When the
moving electron enters a magnetic field, a new component

of force is encountered.



Fm e(v x B) (2.10)

where 7, is the component of force due to the magnetic

field
B is the density of the magnetic field.
The total force on the electron is then

F -ev0 + e(v x B)
(2.11)
-e(70 vx B).

Let us postulate that the above equation may be
written

F -e(VS + 90') -VU, (2,12)
where 0', as well as 0, is a proper potential function.
Note that this is equivalent to assuming that #' exists
such that




Such a potential does extat and is defined by we eg




where IR Tootor velosity.





11


II is the rctZor potential of the magnetic field;


f.e.;., ;B V`i A., Finally, the Lagranglan is applicable and


0ii. takefS~ the form~



;; ; L Isat (eg + er') (2.15)


1 2
- av- eg + e~rv4).
2r


datea


:ii i
E~i~ii:O1CPli, :,,;~errE. eqyation is nor rpplied to the prob-


1~~1::: liMnB r~ir pi E~lindricrl coordinrter. Ileier
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ha

12


I


wiii


pte




tiiB Musert p ps






; I~;- ;;; ;;;.;; ..; g agi~,~i~~








for the B-coordinate,

or 6 -erLg + C (2.18)

and or .the z9- coordinate,

as -ecz + eO z (rAd). (2.19)

:;; Equa~tions 2.17, 2.18, and 2,.19 constitute the de-
eg~ign r-EquirementsB~ for the beam. The utility of these ex-
.;pI~ression IS clarified in the next section.

Interpretation of Mlagnetic Potential
in Terms of Flux

~ c Mant "vctor Potentials are erasly defined math-
gapt~i ally. Thyre not, however, easy to visualize,
]:l;-to go tal they have the direction of currents which esta-

bl 4 M;~~jtteld.P Where currents are absent one may imaagine
Bt*, f~ira~I~ction wich oul have-produced the fiield.
Tipp ageetic. fielda cut by the ~beam in this tube is
tO1~;$'e is'~~b eptabishe b two concenltric pole pieces
3l~r~~~treal@il''iii.ii p..i; The angne~tic. potential is defined by

9 x .A(2. 20)
4 axallygymetrialthe v4Btor jI

84894448~Lji:? Componen~ it:. Assuml~ e~ this is the
Crilill be~aten ed euato 2.20.)





14

6A6
r component E Br "
6z
9 component BO = 0 (2.21)
z component BE 0 (rAg)

The equations 2.21 suggest that Ag If(s). In what follows
r
this is verified, and the particular solution of interest in
this paper is found. Figure 2.4 shows how the flux density

may be expressed in a manner most applicable to this prob-
lem. Clearly, the flux density is given by

B B dY 1 dY (2.22)
r a 2wr dz

where T is the flux traversed by the beam as it progresses
in the transition region. Substituting this result in the
first of the equations 2.21 yields

1 dt &Ag
2wr dz E~

or

dT - 2r--(rAg)ds.
6s

But in view of axial symastry and the third of thy a path

2.21, all derivatives of the quantity (rAg), except is
respect to s, are nero; hence 4.(tty)ds I the tot
eatial, d(rAg). Therefore,



When the bene has posed compipt@ky thy a
has trtrarged the total tha, which sF





is in a~iregion ofX zero magnetic potential. Introducing

this bounary~ condition evaluates C&*


C4 fc*


2wr
$gWa btiturt~e 2.24 in 2.18 to obtain




Befor4r l~~Jtt~ autta oflux begins (i.e.,l 0) the angular
$sl1 is llllto and1







0 .(2.25)

L~IIPI~tisB h q C.5is recognized ase Busch's Th orem
,ii::toopetty an~i;i~~~d ~prosbile~m at hand. It wras found

i.. ~ g, al*p.~~~,j:~. 0.Syetituting this result and






:i;iXI requfre ed that there be no
by 4 boundri~Bs .Callagt these
B aton pggtagthat.









fa $. 2 (2.26)


Equation 2.26 is a most significant one. It speci-
ties the value of thre radial component of electria c.Held art :'

beam boundaries in terms of flux traversed. To complete the. ~''.I

boundary conditions the potential 0 must also be spercified.l ''"
This is accomplished by substituting 2;.25 iso the energ
conservation equation, .':;;.i ri


2

At boundaries the electrons are perrmitted no r~Ladli;: vetootti~Zl:li
therefore




or
(0)ro I (s)2

Equaions2,25 $.B, an $.
the bem desin Cobdtions.Statt
gested by gatit -dfie $A
The deivatiau toad are
give~~ furhe to
pr; .;;;. ..~:~l.!:; I~'ili:~;I;!':i~ ~ i





Act1~pually,, thi~s is not the case. Simply specify (0)ro as a

Isaotiqnof a a nd let (s)ro as a function of a be what it
w;ill;.. 'hi4s procerdure is.possi'ble becauseroiide

peaentof (0,i Certainly an initial velocity in the
axial, gatrioneut be imparted to the beam, and this
WelphBity pas heb~.~~3 malltintaid great enough to keep shot
80196: Subduewu d. Also, if the tube is to be of practical

ve )ebese pas.~t progress axi~ally after emerging from
Bitio regin. The selection of a function for
gu~,~%::y th reasons theriefor re presented in Chapter III.
.eha Md i aus~rt be knoawn to order' to utilize the
Me@ the hlcal trajectory is specified. Equation
.1!il~i~::' th@>vthe o a foltlows readily .from the law

~Z~i~ W~Jiid i;.r shergy. Upon emerging from the transi-



)ai pcfid() e ae()
Illllli1 We found eiallarly at any point:
I~li~llllli;ll~rbtl l thereli~~i. ;amems to be A t le r a




~~Bi~k B~i~~ Lot onur of a npo
~~il!l:~ 94#i .bc~E.i';,ai2 tonbe Afthe all o
11 iil is





18


practice the flux distribution is somewhat like that shownr

in Figure 2.1

In the terminology of preceding mathematical devel-

opment the total flux passing from the inner to the outer

pole section is P Thus a beam segment at point P2 would

have cut Tg~ flux lines, a beam: segment at.~P1 would hav~e

cut F1 and so on. Figure 2.5 shows a typical cyurv o f 1n

versus distance through the transition region.. T;his ifs an ::i~

empirical function and it may be found by flux-plotttagy or;;~;'

analogue methods. One method for doing this is discussed ;;ltjl~

in a Ilter chapter.




; 1 .






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Ilillil oi. "







.1









;I Ii~i












CHAPTER III


QUALITATIVE PRESENTATION OF TUBE DESIGN

Pierce Electron Gun Approach to Cathode Re~gion- .
Analogue Method

This s tudy was -arde simultaneously with ab .selpar~ate ~:;ii%$~~
study dealing with the application of abollow :ylinadrical:~
electron beam in a specoifice traveling-wave3 ,am~plifie~r tuberei;
The amplifier tube .ia itself is related to thig a wk op
insofar as~ specifi~ca~tions of phystEal di~ll~mensions an wed
potenrtials are concrnPef~d.. .It S wasrequf)irbed that %he eam ;,~~l
forming and collimating d~e ice .proiea10 1 1dp

aetler~slr and an out er:' radb ~~ ~ ~ ius pt 1.84 Watime) re
beam.i;;i potential:;: to the edb;e regid a
volts ; ~ii: il i.I!:;i l~~ii libii
be~ Thws sethets i~.~rap d9t4t~~~~~~Esigel
to ~ ~ ~ ~~iZ gpoleyWtf h pqqpp
Bhap98r Bad 998 q, ig .









Application of the electrolytic tank to static
fieldd problems having axial symmetry is accomplished by

tilting the tank through a small angle. See Figure 3.1.
Th~e wedg of electrolyte thus formed simulates a section
ofP ~tha~coplete cylinder. Current flow lines in the elec-

rel~:-QYte create a potential-drop field which satisfies

4lplace's equ~ation as does the true potential field in the
In; be The intersection of the edge of the electrolyte with
thi~i-e ~bottom of the tank creates a line representing the axis
of the system.
the ;conrditions which must be met at both inside and

tieboundary surfaces of the beam are:


- (3.1)


*,' 0 E4/3 (3, 2)

qud~lllljil;]il:("iitte 3I';~I.1 is established in the analogue by

~~Iad inst~iftin strip parallel to the axis at a radi-
pti## he dgeof the beam. Thils assures no cur-
8 r6ion normal1~P to the boundary s (See ele-

Mre .A. .*Con ito 3.2.q is eit~ablished! by

qA:Aetecrade. This is a tral~ And er-

$heer Mii e At wa'~i~'s found that two

::ad ~ipt ededcigsur aces~~ serve ade-





22

necessary might be employed. The cathode end (at a) is

operated at zero potential and the extreme end at any con-
venient potential to represent tube operating level. Shaping

and positioning of the surfaces are then adjusted until one
reads values of potential along the insulating strip in agree-
ment with equation 3.2. Electrode shapes on the inside beam

edge are found similarly, the difference being that conduct-

ing surfaces are erected on the opposite side of the dielec-
tric strip, and the strip is moved to a radius corresponding

to the inner boundary of the beam. Measurements are made

using a potentiometer arrangement so as to insure no current
flow to the probe. The system is scaled such that the sur-

face area of the electrolyte is large compared with the con-

figuration of the analogue. Figure 3.2 depicts a cross-
section view of typical electrodes found to the above de-
scribed manner.


Approach to Magnetic Field Distribution
Flux plotting and analogue methods of locating may
notic field lines are widely described in 1tterstate6,7,10
so only facts pertinent to the design of this tube will

presented here.
Reference to Figure 2.1 shows the general 4 rge
sent of the radial field to the tube, and at@r age

section on



















i.
ii~li

xxi; .;il ii. ;'


xx.::i xii;;;i ; ;i...r

i x M;: ii~i: ;;; ii; ; :;
Ex x M

Ex'


ii iii





iiii'';i i
";; ;; ii


ANGLE


::!:

















.I. ;; I lihilli


TX<'mu








,ii xiiii.;








distribution of the field. Figure 3.3 shows a top viewr of
the arrangement in the tankr for taking magnetic field data.
Notice that this data provides equipotent~ial .lines.
The actual flux distribution is a family of curves orthog- :i
onal to the equipotentials. T~he flux lines are found by
skestch~ing in the perpendicular seat, emaploying the curvi-'i'
linear square principle. When t-he flux plot is found., oner
can correctly assume~ that all flux pa~sses between the ex~f-
treme lines .and thatL an equal amount i~ cooki~~nrd. in eachb'

tube defined by adjacent lines. By~ makingi as tearate pii8:j:/l
writh a large number of' t~ub~e a uniformia ~ipfunct-iaofin der
sus axial. distanes is determined. Figure 2i.3 shows suh
func tion Acet ual exjperimen tal resul ts are:: Pr~e33333~bi:ij en~ 'i Ipt

Chapter IV.

Approach tr Traniting.Reglan Desi 4~1 1
I:t is asseedtht te hlfo4

tShe t~~~rlnansitiont~J re ion ell aitast~elr~1
elctoa crs aiaagp
tialii copnet of;i velow

tion to dtaph a 1








26


and


(g)r e m(z)2r (2.28)
o a g"o e

where ro is either bounding radius of the beam. As men-

tioned earlier, the value of (0)'o as a function of z is

somewhat arbitrary. It is well, at this point, to consider

the restraints on the selection of the function.

1. The function must be continuous all along the

beam.

2. It must permit the beam to emerge with the

proper ratio of tangential to axial velocity.
3. It must lead to a physically realizable

focusing electrode system.

4. Final beam potential is specified.

5. Beam radius must remain constant, .

This last condition was applied in the derivation

of the design equations and must be adhered to. Conditione

2, 4, and 5 together specify the total flut, since fro 4



from 2



and from 2,25






217

Intea qua~~tions theZb subsecript c implies final values upon

emerg in; ro tthE~. transition region, W~ith Final potential
;a d 'i.totl (ju fixed, -the final value of (s)r is fixed.

heide of~the. fusction ( 0)r, is arbitrary to the extent
: When no mant~ter winn~re the beam reaches the final

level, Et eems logicall that conditions 1 and 3

be ~ bet :tstid4.t.ra constantti potential do through-
; anition ~~gaso with the curve extended smoothly
:atkilCftacion back~c in the cathode region.
214lfes the 'hoper~ical solution to Laplace's
theWh~getree space about the beam. For these

upli~'ii fo (0)i is used in this partic-
i;ai'?jt;h p obl~eagi Pigure 3.4 illustrates the

'lue .It .bedua radius to -1 ra.
rvt'Qivee~i of potential at r, r, is

$.MG Egpricl rlelaatin are in-
Q-4hotpoit solutions. Conseider

apae4 radiit greTater than
let th-a orderly ~rara of


anqe Mia r a pote~ntiarl



















i~FJ



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30

where h point spacing.
The net-point equation applicable to cylindrical

coordinates with axial symmetry is:


01 gi a + b c d) g 1 c a), (3.4)

where the symbols have the significance shown in Figure 3.5.

All terms in this equation are known except Oc. Solving

for Og we have:

0 8rl 0 2rl (g 4 gd 2rl-h
c 1 b 0 (3.5)
2rl+h 2rl+h 2rl+h a'

With equation 3.5 potentials for all points at

radius rl may be determined. The author has concluded that,

inasmuch as this is an approximate procedure, it is wise to

plot the results in accordance with a smooth curve drawn
through the average of the points. Some of these results

are presented in Chapter IV. Having determined the line of
potentials at rg, the entire procedure is moved up one step

and potentials are calculated for points at rp Note that
the first point is lost each time, so there is a region be-
hind the line y-y' (see Figure 3.5) for which we have no 4

solution. This le of little consequence, however. We bare

only to fix the electrode system aleag aggipotentials a
cluding this region.













CHlAPTER IV

;iiii;.'; ;' DBBCRIPTION OF EXPILIERIMETS

L~~~ ocatioP of Focusing Elctrodes in Cathode Region
Thbe tuibe designed and tested in these experiments
a ~s'reqired -to deliver a 100 mas bea at a total potential

IQ;. 700 :,i voltats. The inside radius was required to be 1.302
gent etersand the outside radius 1.640 centimeters. A

lil44i 4 i e sca~~i~~d~. lS; was selected and the Ianalogy set up as de-
a the fir:i~ellf section of Chapter III. Plain tap
Use~ as a electrolyte with a small amount of

a~;'~ig~;ii~~i~q Agur proper wetting of surfaces. Electrodes

~bit.g~~twin poised oper lept, and the beamI bound-


.tloor~i. Ate a anatber of trials using

895,it ut cncldedthat two properly
~i~~ialld don:ey figure) 4.1l dpic~kts the

? W et t~tory~-run S;hopp dotted




Pt.i;:~~ ~p+rbatheticl
; ~ ~ a tieri;"iil~r; r.i,~;















8l~


M ; 6 B .





.::


:: ; ??:II


i::


'

CI.

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.


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;'iii;;ll3ii;: 'IIEii:ii";;illllliiii;illllllii;illili 'Illii;ilillllli;iil1111
:i
ii' .ilii;ii;;;; ;iii;......:iiii;....
'ii ii.ii;...;liliiiill:''iili. iiiiiiii'iiii;iii;iii!
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:ii.i:.
iIiiiiiiiiii

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il: ;; .;
:i;


pl f: 5f S cn r o to o cc rr




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...........0
Ow eYOcencoOneo00
to a M W C o to t o a C
15i # e-- e ca ts vn I a e U


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go- M W C4 0 a g W c
:! e-- a to a m an a




at ~ ~ e ( n o e at co to rrn 49




ru to SW A to a rD CM In (D t


404cocooec


go 1D at at t @ O t* 48 C
,':ru iF C4 Co~ cm Vo c o u, 0*-





rl CV(* m *t I ,
'; Some;
i'ei xqssem-

ii~;iiiliii i flilii cllrxill;lxllj iii:;;';;I; :"i;
;.iir;..i xx a o
an ag
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/li
i






34


Potential values are shown on Figure 4.1 for the

most desirable runs. Note that at both beam boundaries

the radial component of electric field is very nearly zero.

Where it has a non-zero value it is directed away from the

beam. This insures focusing within the bounding radii.

Fabrication of actual tube electrodes is discussed

in a later section.

Determination of Magnetic Flux Distribution

The method for taking magnetic field data is dis-

cussed in Chapter III. Figure 4.2 shows the results of

data taken in this manner, and Figure 4.3 depicts the flux

as a function of nxial distance. In the actual tube the

total flux employed was 6.2 x 10-6 webers, an amount neces-

sary to give the beam the required spiral pitch.
In order to be sure that the tube was operating at

the required flux value it was necessary to obtain data

relating flux and solenoid current prior to the tube-test.

To accomplish this, the magnetic circuit was completely

assembled and a magnetisation curve taken, netSuring flux

density at the mean radius of the beam. F1st density *As

measured directly veing a gause-meter manufactured by th A

Dyna-habs Corporation. At the center of the pole attgg p
the calculated value of required tha 44ksity was OS up

This corresponds to A.total flax of 6.2'% 10" hats


















t7.r .. I


tI



15
i
a :'iii3iillill
;k;
i:: I
~ P .....I~ if i' .8 S...I il ';:, ;
AE al gitaie t aiiied
is a;li .r. nzannesii
ili ai






.~9 IMi II .1l I a






;'"':;;; 3 7

as. t;1.iC.;~Tnain for.190 milliamperes in the solenoid.
The l~qoatio of the pole pieces with respect to the
too...' :lreg.. o.. .~~i~F~~n wars slomewht arbitrary. They had to be far

ough'-~ awaya toeep, flux out ofI the cathode region, but at
sa imer as niear as practicable in order to shorten

I';i:!'11~, ~~~~i, 1 ; all 1 clima tion length The relative posi t ions
ecte areshowni on Figure 4.2 where the 720-volt elec-
A(t~i/'ii.~r~~e represent the ex ten t of the c athod e regi on .

CllllllijElllFi;: ;Loqil~ :ti~itagl~~ El~catrodes in the Transrition Region
Illnre; r, f~~;i,,~ll~8~~ingag thlocation of electfrodes in the transition
consisted ofIl aryag ot the net point procedure de-
k;hptr ~jIII and d~irawing equipotential surfaces
@#et entfe rgioni. The pole elements consti-
Egge of ze~ro potential~ and had to be shielded by


Iq~::at~ a o te tub a ultitu~de of potential

j~i~~;~lorrected,~~ and plotted in the neigh-
adry, Thei~se calculations are not
g:et: bt fo clrityr a spaple cal-

St ~ ~ ~ ;. hapoi8; dpe ae
I iieiil; the onil or:;ii~lil:jlli beam..il.


~b A gr n ~iin La =n




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ii


e h 7a
81i 720 +
m wro3 'O




, @ is potential at the ith point

Pi is flrux traversed at this point


r 169 x 0-2 ete8rs


hi,iii; -i,;i ...;;; i05 x 10-2 meters;






.i':.g 720 4.6 xr 10 Ti


Ati;~; C poijnts~ 1,. 0, rand 2 respectively we find from








.05lBC~~ 2 10 weblters


41~l~'%i&;:':~~ fO. 10 ebira.




pbaii potenttials are:


~ir;~~'i:~O~d fr~r rqurtian 3.5.

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Calculations for values inside the beam were made

similarly except that h was a negative increment of radius.

In all, about four hundred potential values were deter-

mined. Figure 4.6 is a scale drawing showing the resul~ting

net-point plot. The equipotential lines shown are those.
selected for focusing-electrode design. Notice that these

lines become parallel to the beam after passing into the

flux-free region. Th is mea ns t hat i n th e dr i ft- tubie r~e gio4 .Ij.;!:,I;

the beam may be held in focus by a pirr of coax~ia cylinderp,~;~/
one inside and one outside the electron stream.

Description of Hfollow ~B~em Electroni i0un

and' Collimatin g DQtevice w-;;iI!~llr:i; ;;:r.ll~

Figure 4.7 shows a scale drpawingof te tbe a.0,
actually constructed, rand(figr~e 4.8ia htga, th
tube and test set-up. Th ;ildsac ,in :'is: a t~ ~iii,ll

the important points'a ele~Ctroded sortgaes are 'gty
Axis to pobint D1 .473 inches ti, 5to p








i. : i:gll;i ii:ii


i /



iii'
iI';i"

;i;"iiiiiiiii
.;;iii;iI

;..,;,;; i.i iiiiii i
ii ;iii;ii...


I Ceramicr


G OilC capgper



~Q Steeli Nr 7


AI;iiii;l;

: x


:L'"









This list serves to define slopes of the equipoten-

tial surfaces required by design calculations.

The angnetic circuit was of cold-rolled steel with

niket plating. The magnetic winding was sealed in a

pBartially evacuated glass envelope to isolate it from the

hjigh~ acuumn.
The cathode cup and back plate were pressed sepa-

4D~iCratel fcro thin nickel sheet using special dies. One end

,llof~!i;,;I~b the heat~e~r element was spot welded to the back plate and

thl!:l;,l;;:i~~~e other end passed through a hole in the plate. The cup

iwas theni preipacred for cathode coating using the C-1A proc-

eas,!1 'i!~ 1~1~~~ ?Eh~e oxide coa~tin~g used for this cathode was pre-

"ar by the Sylvania~ Com~pany. Prior to application the

arptpr ~at innpension, was rolled slowly for about two
,,,,;,iieiiii, be-htho~~~de cup and back plate were then welded to-

c###2 p'~~[leded on;o a ILthe LFo coating. The coating was
sli:4niiaki g&"~ the assembly~ against a fine cmel-hair

RA pienioe were mad~e, allowinE each coat

($&Bp.Finally the surface wpas scraped,


trorthis c..~a~bthode was made by winding

aiiii. i; oni.;i;S:~ii i; ecial igE with twiin-a tral graoves
better Eq~~' has fired ad co~ated by .the

6atibAin hei laoraoris.This






44


The ceramic parts shown in Figure 4.7 were machined

from Lava, Grade 1137, furnished by the American Lava Corp-

oration. This is a natural stone particularly suitable

for high vacuum work. The electrode surfaces were created

by painting the appropriate areas with Troy Burnish Silver
Paste obtained from Hanovia Chemical and Manufacturing

Company. The ceramic parts were hydrogen fired prior to
silver-coating. The coating was then applied and cured

in a slightly oxidizing atmosphere at 7500C, after which

the pieces were again fired in a slightly reducing atmos-

phere.
Prior to hardening, minute holes were drilled in

the ceramic parts to permit the passage of electrical

leads. These holes terminate in the planes of the elec-

trode surfaces. Leads of .010" OFHC copper were passed

through these holes and attached to the silver coating

using a very small plug of solder with high tin content.
The solder was then melted with a clean iron creating a

bond with the silver, and at the same time leaving the

electrode surface perfectly flat,

Colliantion of the beam was checked by sensuring

current to the beam Collector, collector rings, Red electr

surfaces. These tube elements are ident4tied on Piture 4.

According to Table 2. The beam collector was asc$1444)fy

nickel tubing and the collector rings were set he 914,







45


Tube-Test Procedure and Results
The tulbe was tested in a demountable chamber evac-

natied by a three-stage, glass, oil-diffusion pump manu-
faCkgi~: 1~~~redb Distillation Products Incorporated. The glass

stea~i!ncludede a cold-trarp consisting of a pyrex finger
;Cf inch -fn diameter and about ten inches long. The metal

b!,il:;: me~l;ii~~ip, l~ate was sealed to the pump with Apiezon "'W" wax,

C:,,Progie~~.T.aue were measured using an RCA 1949 ionization gage.

Aftr mre~ than twenty-four hours of continuous pumping a

~jill'presure o rf about 5 x 10-5 ma. of mercury wa~s obtained.

A dewr conainin dry ice and acetone was placed on the

- Sold .4tap n the system pressure dropped to 10-6 m.o

4 tlll;,~i~"""""";islhis point ac rtivation of the cathode wars attempted

ar, gsui~de., the procedure outlined in the MIT Tube


6 rapid evoution of gas ceased at a heater power

sl~llill~l ttslill,~li~ As;; all:;~.D. PC. voltage was then applied
p~Z~lllll:ilEIle a agli;.; eettejde system, and current flow was

penyiteQd by an fincrelase in pressure due
ana~~ii~lJ;l. ICurrenXIt was main tained un til1

804iiiji~~~i; i :. stable. T his process wras con-
eaii~j~ii~:i W 'te operating at rated vol tage .



the t 4, potential ere~ applied
ha the-agnette cicuit was








































sit


eneC~

the

s ho~

da t


rgized. The following table is introduced to enable
reader to mnake cross-reference between tube elements

wn on Figure 4.7 and Table 3 which presents tube-test


TABLE 2

TUBE ELrIEMENT 1DENTIFICATION

Beam current collector- Bl~emet 2

Rinag collector Elementa 3 an~d 4~b

Inside accelerator: ~ Surf~ace~ J

Outsitde acel~erator- -: ;S~urf~ace V

Focus 732- Sufe I

Focus 710 Su.;. "lrface TUl

Focus 8 100 - Surthee; :;USa~c


Pocus 15 Settan

ii Io ostryi

he~l acts t lectr
es i corent hrobhe


-to 144

&@*44 91


















>*re t rl a -








re r e r t r






aM M rQe a a a to O
0- a In t

s~I we C onc


rI lb to In r




rl> Il t rk a 1





ii I( r l a I


eYt t* W rr W







W e e 96l llllll; ''I'i . ~ C cI
M M M MMlllli
i'O d d d
OP~ *4D a a a
"49. 0 0 i 0 0I





dIII1IIII~lll 1 illClliiigl~ i~i pm ;' ;i; i C c aC
:rt `k t* M to *D


s:g i i Q O
Illlllllllililllill0 OU
p r- m ED r






48



o a 0 o e a



0 0 0 0 0 4



O O O O 0 0



O O O O O 4
N M N N


O O O 0 O



C O O O O 0 0

O
O



A 0 o a 0 a a


0 0 O O 0


e a M
M < 4
M < M
& M 0 0
M 0 > >
o > e
e a o a
M 0 0
09 A OW We a
CV d@ >@ M MP
0 SD 00 CO 90 e
> > > Q> @>
MM









none of the current would be attracted to the existing

parts of the accelerating system. The author believes

greater percentage through the gap could be realized by

gi~insg th~e surfaces KL and WIX (see Figure 4.7) a greater

slopel oir by operating them at a slightly negative poten-
tfat1. Either ;procedure would establish a radial component

of electric field away from the beam edges. This behavior

a~rs noted in the analog design of these surfaesp. It was

at ossbleto operate them at negative potential during
"the test because they were connected to the cathode inter-


Cogination within the cathode region proved to be
qutt saisfctoy. nstruments connected in the leads

to ~ ~~ th-scsngeectrodes indicated zero current, or cur-

il eatLl@i "l!:.i'if rso~; alfgight as to be negl igib le The measure of beam

ll:l;light:~~iIan b the transit t ion region is made by compare ing
beiiiiiiiiiillii: iai-lllllllIi';ll~j~~~carre..i.t w;;;ithrncret As depicted in Figure 4.7,

rp8t~eT: systemt cosite of a sher-shaped beam
ri'~~,j~~iii~:~: wItWej two Trin-col lectors placed one-tenth of an

~~li~ therllllong the axis.1 T~he beam collector was located

.heyond the i:~~~floal 'focusing~ elctrode. Per cent
li'iris.Xeined:~3 asr tIhs raitio.of beam currdht to
enty, ;ii~Cting;the tansitio regon times one

is Pe dlel A:evau Ood measure of colli-

.ti as a ed he ring dytaS wascsiderably






50


4.47 square centimeters and the latter 3.14 square centi-

meters. Table 4 presents collimation data for the test.

Note that runs 4 through 8 clearly indicate the

degree of beam control available with the small radial
field. For these runs beam potential was constant, fo-

cusing electrode potentials were held constant, and mag-

netic field values were varied. Run 5 indicates that for

low flux density a considerable amount of current flowed

to the inside ring collector. Strengthening the field

gave a greater angular velocity to the constant energy
electrons and increased the beam radius. This seems to be

confirmed by the data observed in runs 6 and 7. Design

information indicates that optimum collimation should occur

with a pole-Center flux density of 59 gause. The data show

best results for a density in the neighborhood of 50 gauss.

It follows that as the flux was increased the beam radius

became too large and a greater percentage of current flowed

to the outer ring.






51





TABLE 4
COLLIMATION DATA
Flux Beam Ring
Per cent
Run Density Current Current
C omn
WS'"' gauss ma ma

I 58 D 0 ----
2 5 8 4 6 4 0%

3 40 6 11 3 5 3%
4 40 8 14 36.4%
5 35 4 18 22 2%

6 4 5 1 3 11 5 4 1%
7 50 17 6 73 9%

S 5 1 6 8 6 6 6%
9 SO 23 12 65.7%




52


FIG. 4.8 -- THE ELECTRON TUBE AND TEST SET-UP














CHAPTER V


SUMMARY

In this work it has been demonstrated that a rela-

tively dense hollow electron beam may be formed and con-

troll~ed without the use of a strong axial magnetic field.

'Pierce's9 electron-gun technique proved to be quite success-

. In1 in the cathode region design. This was to be expected,

s~~ince all values wrere held within the known physical limi-
tat~~ii a of such beams. For instance, the perveance (de-

fia :~. as urren .divided by voltage to the 3/2 power-) was
ril.:8~b- 10"' raseres per volt3/2, a very moderate figure.

Anohe lC~~~i imitation in establishing dense beams is

the -formation of a irtual cathode somewhere along$ the
launn.. When;; thei curent ir increased indefinitely, the

R"""...;ixi;'l;d;d~~ o;`.:~~g3tf poetal wi thin the beam may be so grea t

,ii se he ielocity o,...;.~ ii4p~t f the inner electrons to zero,

a 16 it~ is assume that the heam boundaries remain

.,;I eaul th dur~en is limited; dA equation
iAlbidkl hajs fxed the limiting value at
apparest his would permit a current

shediscuss '1 thePreBSien~t work. Since






54

100 ma was the greatest value expected, the tube operated
with a more than 500 percent margin on this limitation.

Chapter III presents a very workable method for
finding focusing surfaces in the magnetic flux region.
However, obtaining the required flux density in practice

developed into a considerable problem, the difficulty

being that the total flux necessary was so very small. It
was required that the beam emerge from the transition re-

gion with an energy of 720 electron volts and with a ratio
of tangential velocity to axial velocity of 1.55. (This

requirement was based upon a practical pitch for a wound-
helix, slow-wave guiding structure. The wave guide was to

be wound on a ceramic core and operated within the hollow

beam.) The above figures led to a tangential electron
velocity of 1.34 x 107 meters per second. Substituting

this velocity and a radius of 1.302 x 10-2 meters (the

inner beam radius) in equation 2.25 gave 6.22 x 10^
webers for the total flux. The resulting flux density at
the center of the gap was found to be 59 gausa. This low

density was difficult to obtain and difficult to measure 5
accurately. However, performance of the tube when sub-

jected to various flux densities in the neighborbood of

59 ganes definitely demonstrAted that (pan collisation a
controlled by the weak radial field,

No good Method for checking the pitch (be .#,pt
trajectory of thebean as devised. Conalder4gge qught






55


was given to the possibility of getting a visible image of
the beam on a screen coated with willemite. Had this been

done, the cathode might have been marked causing a shadow

to appear on the screen. The angular displacement of the
shadow from the cathode scar would have furnished a measure

of beam rotation. The author was advised against this

technique, however, because of the relatively immense beam

current.

It should be mentioned here that the design pro-

cedure deals only with the surface layers of the beam.

Conditions for equilibrium of electrons at the inner and

outer radii are presented. It is assumed that the trajec-

tories do not cross radially, hence none of the internal

electrons reach the surface boundaries with radial velocity

components. This assumption may not be entirely correct,
but there seems to be sufficient experimental justification

for ite usel5

Figure 4.6 shows that the equipotential surfaces
in the Laplace field region become coaxial cylinders after

the beam emerges from the magnetic flux. Thus, in the

drift-tube (traveling-wave section) space it should be

pos@tble to hold the electrons in collimation by & pair of
coaxial electrodes, one within and one without the beam.

The ouod-helix referred to earlier might serve as the

$nner elettrade if operated at a suitable D.C. potential.









Harris has presentedl5 some experimental evidence supporting

the above conjecture. This was not the primary purpose of

his work, however, and further experimental work on this

type of drift-tube would seem to be a very worthwhile

project. The author is presently engaged in a continuation

of the research reported here and hopes to gather consider-

able drift-tube data during the experimental phase of his

s tud y.









.1 ;i













APPENDIX I

THE NET-POINT EQUATION

Equation 3.4 results from a consideration of the
difference form of Laplace's Equation in cylindrical coor-
dinates. For the axially symmetric potential field sur-
rounding the beam, Laplace's equation is:

42# 420 4 1 60 0 .
6r2 6z2 7 NE

Consider Figure 3.5. At point 1 the partial derivative of

potential with respect to r is given approximately by the
relation:



(60 Sc Ma
br 2h
1
The second partial derivative with respect to r at this
polot te:


(62p 1. We 1 91 b
1 h a


: 1 Od 1 91 b
1622 b h
1

57






58

Substituting these values in Laplace's equation we obtain


1 Oc- 1 1 1 0 011 E 4- 11 & O .
h h h h h h cl L **

Collecting terms and solving for 01 yields the net-point

equation,


I a b c d) b-(0c Sa)- (3.4)
1

The method of application of the above equation

depends upon the available boundary information. The value
of the potential at a point in a charge-free space is the

average of the potentials of all points equidistant from

it. Then if the potential of a sufficient number of bound-

ary points is known, one may make a judicial guess at values
for the remainder of the net-points. The assigned values

may then be used to calculate a more nearly correct set of

potentials. This process, which is known as the relaxation
12
method, should be repeated several times until further

repetitions yield no change. That the procedure is a con-

vergent one has been definitely established. See, for e
instance, reference 13.

The application of this method in the present wQgg
is rendered easier by a knowledge of the electric field

intensity as well as potential at the bees boundary. With
this information it disposable to calculate talkep at '






59


points immediately ad jacent to the boundary leaving only
one value unknown in equation 3.4. This procedure is

discussed in the section on Approach to Transition Region

Design.













APPENDIX II

ON THE ELECTROLYTIC TANK ANALOGY

The analytical solution to Laplace field problems

is limited to cases having only the simplest boundaries.

In applications where the boundaries cannot easily be

expressed mathematically solutions become difficult or

impossible. Solutions to such problems are often found by
making use of the electrolytic tank analogue. This method
was first used by Fortescuel6 in 1913 and has since become

a very familiar scientific tool, Many complete treatments

of the theory and application of the tank are to be found
in literature. The reader is referred, for instance, to

Volume I of Electromagnetic Fields by Weber10 or to Elec*

tron Optics by Coslettl7. These remarks are included as an
immediate reference and as an extension of the discussion

in Chapter III.

Consider a pair of electrodes in free apace with a
source of electric potential connected between them. Con" 9

sider, also, a similar system With the electrodes immerged

in a homogeneous, alightly-conducting liquid. To the

former case lines of electric flux perseate the space, fadf









in the latter case current flow lines pass through the

electrolyte. In both cases electric fields exist in the

regSous about the electrode arrangements. These are vec-

tor fields and are, by definition, proportional to the

gradfsient of potential. It is now shown that each of the

potential functions satisfies Laplace's equation.


Potential in the Electrostatic Field
Around the Electrodes

It is presumed that no charge exists in the region

.aBout the electrodes ini space. By definition


.~ E = 90, (II-1)

wherre E toe the electric field and a is the potential.

ii4)::4A 8 electric f.Lux line must originate or terminate

on charge andinnauch as charge is absent, this field has

noil~iii 194rgba iiii;;. li;I;.!i .se Bpesed mathematically, this states that











D.s~~~:. -~~"n (II-3)






82


Potential in the Electrolyte Around

the Electrodes

The electrolyte permits current to flow. At any

point within the liquid the current density is proportional
to the electric field and is given by


J yE, (II-4)

where J is current density and y is conductivity. The

current density vector obeys the equation of continuity,

V-J by
bt
where p is volume charge density. Since as much charge

flows out of any volume element as flows into it, the

partial derivative of p with respect to time is zero, and
the divergence of the current density is zero.

V*J 0. (II.g)

Substituting equation II-4 into II-6 yields the following:




Again E is the negative of the gradient of a poke
tial field, It is significant that, in this instance, t

difference of potential from point to point la created b

an IR drop. Let this potential be desktnated by thyaya

V. Thus E = -VV and equation 11-7 becoagg






63


or

92V 0, (11-8)


Equation II-8 is Laplace's equation, and the IR

drop field in the electrolyte is seen to be exactly analo-

gous to the electrostatic potential field. The importance
of all this lies in the fact that the IR drop can be meas-

ured whereas the other potential cannot. A description of

a typical electrolytic tank application may be found in

Chapter III.













APPENDIX III

ON FLUX PLOTTING

Besides the electrolytic tank analogue another

good way of making a field study is the graphical flux plot.
A magnetic field is chosen to illustrate the method.
Refer to Figure III-1 which depicts a flux tube
within the field. The tube is bounded such that its walls

inclose the same total flux, AY, from beginning to end.
The surfaces normal to the flux direction represent mag-
netic equipotentials, Ug, U2, and US. Assume that there
are no variations with respect to x, and that pacing is
chosen to require equal potential differences between adja-
cent surfaces. Let the following table of symbols apply.

H Magnetic field intensity
B Magnetic flux density

a Permeability
As Spacing of equipotential surfaces
Ad Reight of flux tube
Now the potentials 04 and Ug as be expressed we
follows:













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66

and


U2 U1 + 91681. (III-2)

Subtracting equation III-2 from III-1 yields

U3 U2 U2 Ul + H2As2 HIAs1. (III-3)

But since Ug U U2 U it is seen that


81681 H26s2. (III-4)

Let H now be expressed in terms of the flux within the
tube.

H 1 B 1 AT (III-5)

Substituting this in equation IIT-4 yields


As1 as2
(III-6)
ET{ 612

Thus, spacing of equipotentials is also a mpasure
of the height of the flux tube. Then it families of orthog-
onal curves are drawn in the y-z plane sweb that they fore
curvilinear squares in that plane and shttafy bonad4ty ten
editions of the potential, the reso1t is a plot of On W
lines and equipatential liate, The latter gpderate 0
faces when transtated parallel ## the XA$xie.
The above discussion is for two-di eyeleg






67


having axial symmetry. The graphical solution to any field

problem is rendered easier by making use of all available
information. For instance, equipotential surfaces are

parallel to boundary potentials near the boundaries. Also,

quite frequently, either flow lines or equipotential lines
will follow lines.of symmetry.

The magnetic field problem encountered in this

work was that of finding flux distribution. Magnetic po-

tential distribution was not needed. However, in order to

reduce the tedious work of flux plotting to a minimum, a

model of the magnetic system was set up in the electrolytic

tank and data providing equipotential lines were taken.

Then there remained only to sketch in the orthogonal family

of flux lines. Thus it is sometimes possible (and desir-

able) to combine these methods of field solution.












BIBLIOGRAPHY


1. Samuel, A. L., Proc. I.R.E., vol. 37, pp. 1252-58;
November 1949.

2. Wang, C. C., Proc. I.R.E., vol. 38, pp. 135-147;
February 1950.

3, Pierce, J. R., Jour. Appl. Phys., vol. 10, pp. 715-724
October 1939.

4. Wax, N,, Jour. Appl. Phys., vol. 20, pp. 242-248;
March 1949.

5. Harris, L. A., Proc. I.R.E., vol. 40, pp. 700-708; 195

6. Pierce, J. R., Theory and Design of Electron Beams, D.
Van Nostrand Coi KEY, TTEC TERFliBRE 10290.

7. Spangenberg, K. R., Vacuum Tubes, McGraw-Hill Book
Company, New York, 1918.
8. Killer, L. R., The Physics of Electron Tubes, McGr wR
Book Company, NEWTKr1EF"BRER

9. Goldstein, Herbert, Classical Hechanics, Addison-Wea
Press, Inc., Cambridi ~10015000EE GEETTs, 1951.

10. Weber, Ernst, Electromagnetic Fields, vol. I. John
& sons, Inc., 8 THETTISUT-

11, Tube Laboratory Manual, Research Laboratory of 81*
ACTITCITEE5FTHE ITEEsachusetts, 1951.

12. Southwell, R. V., Relazation Methode in 7%eoreties
Physics, Oxford UnI FEITY PF 5 7 EEglEERT"TO $?"
13. Batenan, H., Partial itterential Equations of Nat
ematical Phya

14. Ca b ck C J. u n


68







69


15. Harris, L. A., Axially Symmetric Electron Beam and
Magnetic Field
sachusetts, 1950.

16. Fortescue, C. L. and Parnsworth, S. W., Trans. A.I.E.E.,
vol, 32, p. 893; 1913.

17. Cosslett, V. E., Electron Optics, Oxford at the
Clarendon Press,













BIOGRAPHY

Leonard Caldwell Adams was born in Saluda, South
Carolina on November 7, 1921. He began his undergraduate

program at Clemson College in Clemson, South Carolina in
1939 and received the degree of Bachelor of Electrical

Engineering in 1943.
In 1950 he received the degree of Master of Science
in Electrical Engineering from the Oklahoma Institute of
Technology at Oklahoma A. & M. College. In the summer of
1951 Mr. Adams began work leading to the degree of Doctor
of Philosophy with a major in electrical engineering at
the University of Florida.
From 1943 to September of 1946 Mr. Adams served in
the AU.S. At the time of separation he was an officer in
the Signal Corps. Since 1946 he has been on the staff of
the Electrical Engineering Department at Cleason College

except for various leaves of absence during which he parad@d
graduate work at Oklahoma A. & M. and at the Univeratty of
Florida. During these graduate programs Mr. Adams held
teaching and researDh asetatant positiqD8.

He is a member of to Bett Pi, Phi Kappa Ph(, 2124

Kappa Nu, Sigas 11, sad The Institute of Radio Bagisegre/

70




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