ON THE DISTRIBUTION FUNCTION OF SURFACE CHARGE DENSITY WITH RESPECT TO SURFACE CURVATURE

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ON THE DISTRIBUTION FUNCTION OF SURFACE CHARGE DENSITY WITH RESPECT TO SURFACE

CURVATURE

L. Enze

To cite this version:

L. Enze. ON THE DISTRIBUTION FUNCTION OF SURFACE CHARGE DENSITY WITH RE-

SPECT TO SURFACE CURVATURE. Journal de Physique Colloques, 1989, 50 (C8), pp.C8-41-C8-

46. �10.1051/jphyscol:1989808�. �jpa-00229906�

COLLOQUE DE PHYSIQUE

Colloque 68, suppl6ment au n o ^11, Tome 50, novembre 1989

ON THE DISTRIBUTION FUNCTION OF SURFACE CHARGE DENSITY WITH RESPECT TO SURFACE CURVATURE

L. ENZE

Xidian University, Xi'an, China

Abstract I t is known exprimxitally that there e x i s t s a functional relationship hetween surface charge density a and surface curvature k on a charged conductor. However, t h i s relationship has never been found i n electrostatics up t o now. In previous papers /1/

and /2/ we have discussed t h i s problem and derived the distribution function of 0 with respect t o k. A supplemntary elucidation is given in this

paper

t o explain why this distribution function can be established firmly.

1

-

HISTORICAL BACGKDD

In 1747, B.Franklin researched

the

t i p effect and was convinced t h a t the larger the surface curvature and the more projecting the part of t h e surface on the canductor, the greater the distributed charge density /4/. Up t o 1963, R.P.Feynman asserted, /5/

For k>>O,

a

^.s k

In 1828, G.Green expressed Laplace equation i n the form (1). H e solved it and obtained the result, /3/

For k=O,

.

⁼

^-

^(I+&)

where An is the path difference along the flux line, AV the p.d. over &I and

e

^the d i e l e c t r i c constant. /6/.

In 1830-1839, M.Faraday undertook investigation by experiment and showed t h a t the more negative the k and the more depressive the part of the surface, the smaller is a. Now w e recognise approximately /7/

For k<<O, a = l / k

A l l these results were ascertained by i n n a r a b l e experiments. We can show them synthetically i n figure 1. Obviously, i f other conditions are

the

sam, the distribution function of o with respect t o k is a nonlinear function instead of a linear one. As k increases from

-

= t o

+

^{w, (T} changes frcm 0 t o f

-

monotonically. ^Our purpose is t o find t h i s nonlinear function which is based on our forerunners1 work.

2

-

THEORETICAL BASIS

The derivation of such a function was given i n previous papers /1/ and /2/. H e r e we derive it again in an easy way.

I t is known that the Laplace equation can be written a s

where E is the f i e l d intensity a t a point in the f i e l d and the differentiation dE/dn is with respect t o the outward normal of

the

equipotential surface with average curvature k a t t h a t point. (1) was established by George Green i n 1828 /3/. ^{A s} it is one of the more inportant equations in electrostatics, we w i l l henceforth refer t o it a s Green's equation. ^hl? can express it as

dE/E

+

^{2k (n) dn=O (2)}

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1989808

Integrate (2) and (3) along the force l i n e from a point with f i e l d intensity Eo t o a point with f i e l d intensity E, where the path increment is bn and ^the potential increment is AV.

Hence we have

substituting (4) into (5) we get

Set

the man value of the fundion k(V) from 0 t o AV and t h a t of the fundion k (n) fram 0 t o An respectively, then (6) can be written as

This i s a general and accurate f o m l a t o calculate the f i e l d intensity a t a point on the equipotential surface with high curvature. ^Hence

It is a general and accurate fonnula t o calculate the surface charge density a t a point on a conductor surface with high curvature.

3

-

THE DISTRIBUTION FUNCTION OF 0 WITH RESPECT TO k.

For h < < l / k ,

o = 2EkAV

exp (-2kh)

-

¹

,

(An<<l/k)

All variables of o (k) such a s El k, An and AV are measurable, s o 0 can be determined d i r e c t l y . A . An can be selected a s a very small quantity, even when =roaching zero, k can be varied in a very wide range, even from

- -

^{t o}

⁺

^{0 , and 0 can be} obtained with very high accuracy. (11) is t h e d i s t r i b u t i o n function of 0 with respect t o k which we want t o find.

Equation (11) can be expanded i n t o a s e r i e s form,

2EMV 1

= exp (-2&)

-

1 (12)

~f we let An be a very small constant i n (2), then f o r d i f f e r e n t values of AV, we can draw t h e different d i s t r i b u t i o n curves of ^(T with respect t o k a s shown in f i g u r e 2.

r e s u l t r e s u l t

1

0 > k

k < < O 0'k k > > O k < < O kZ0 k > > O

W l / k a=- EAV ^{a= 1} /k a=- EAV oak

-( l+kAn) oak ---( ltk4n)

An An

Figure 1. Forerunnersf r e s u l t s Figure 2. The d i s t r i b u t i o n curve of a against k

For k>O, the larger t h e k, the larger the d i s t r i b u t e d 0.

For k>>O,

o -

^k. This is the r e s u l t of Franklin and Feynman.

For k+ 00,

lo1 + -.

^{( ~ t} is inpossible f o r a r e a l conciuctor)

.

For k 4 , o =

-

E AV(Z+kh) /h. This is the r e s u l t of Green.

For k=O, o =

-

E A V / h .

For k<<O, _{For k 4}

_{- -,}

o

-

_{0 4 .} ^l/ko. _{This is the} r e s u l t of Faraday.

For b + O , o =

-

^{E dV/dn.} This is the f o m l a obtained from Gauss law t o calculate 0.

A l l t h e deductions from the d i s t r i b u t i o n function (71) are in agreement with the experimental r e s u l t s and the conclusions obtained by our forerunners. Hence under the r e s t r i c t i o n h < < l / k , (11) is just the d i s t r i b u t i o n function of surface charge w i t h respect t o surface curvature which we are looking for.

We enphasize t h a t (11) is correct only under the condition that h < < r (=l/k)

.

I f w e a ~ l y (11) f o r engineering purpose, An may be too small f o r measumwx-it. Hence ⁽¹¹⁾ is only f o r t h e o r e t i c a l use. For engineering use, the general and precise expression (8) o r (6) w i t h o u t t h e r e s t r i c t i o n i s available.

~s

a

check on formla (81, some examples were

given in

paper /2/. Here we give another one.

Exarrple, suppose we

have

two infinitely long

line

charges of site sign

a,

and a distance 2a apart.

We

choose our origin 0 halfway between t%

and -%

^{the x}

^-

^{y plane}

m d i c u l a r to

charge

lines,

and the

x-axis goes from +A to

-A.

^Find the field on

the

x-axis (Fig.3)

.

Solution: The potential due to

both

charges is the sprposition of

the

potential of the two lines:

[y' + (a+x)

'

I [y' +(a-x)' I Equipotential lines

are

then:

+

(a

+ X)2 = exp(-4mvA) = B (14) y

+

^(a

-

^x)2

-a +a

where B is a constant on

an

equipotential

line. This relation is rewritten by: Figure 3 Details of example

-

which we recognize as circles of radius r = 2aJ B/(l-B) with centres at y = 0, x = a(1

+

^{B) / (B}

-

^I),

then the average curvature of equipotential surfaces

are:

Hence, on the x-axis

I"

^{2k (V) dV}

This result coincides with'the result gained by using E =

-

^dV/dn.

A~WUC$ the result gained by using fonrmla (8) is more cmplicated, it shows that f o m l a (8) is indeed correct.

5

-

APPLICATIW OF THE FORWL?i

The general formula (6) o r (8) when supplemented by data masured f r m experiment using an analogue technique can be used t o calculate

the

f i e l d intensity a t any point i n the f i e l d accurately. /2/

The fabrication of a f i e l d emission cathode is a very interesting problem in modern electronics technology. In order t o make

the

surface of the conductor produce f i e l d electron emission, the f i e l d intensity on the surface of the conductor ^must be over 107 V

&.

According t o formula (6) o r (8) we know that in order t o a t t a i n such ^high intensities, the potential difference AV between the electrodes should be enhanced. However, i n order t o enhance ^the s t a b i l i t y of emission and

the

l i f e of the cathode, the lower the AV the better.

For the sake of getting high intensity a t a lower potential difference, we must fabricate the t i p of the cathode t o a high curvature (i.e., increase k) and fabricate the gate electrode a s near t o the t i p a s possible (i.e. decrease An). Wecan fabricate such low potential

difference f i e l d emission cathodes by thin film technology (now known a s the Spindt device).

However, t o calculate the f i e l d intensity on the t i p and in the space is a very c a p l e x electrostatic boundary value problem.

For t h i s purpose, we must use the transformation formula:

where

p

is the geometrical factor (or the transformation coefficient) w i t h dimensions of inverse length. The value of

p

is different f o r different points of the field. The method used t o determine the value of

p

is a very complicated problem i n electrostatics.

Usually, people use

f o r transformation formula, which means t o take

But a s we know, it is only t r u e f o r the e q u i p t e n t i a l surface with k = 0, i . e . only true for a f l a t one. (See figure 4)

.

( a ) k = 0 (b) k t 0

Figure 4. The geometrical factors of Field

For a cunred e q u i p t e n t i a l surface especially f o r a highly curved one, {21) k x c a n s t o t a l l y invalid.

C8-46

However, if we use

to calculate the field intensity El we can get the precise result in principle. Hence (22) is a general foxmula to calculate

P

for any point in the field

with

arbitrary boundaries. It preserves its form even if the boundaries are changed.

/1/ Luo, Enze, J. Phys.

D.

Appl. Phys.

2

(1986) 1-6.

/2/ Luo, Enze, J. Phys. D. -1. Phys.

2

(1987) 1609-1615.

/3/ George Green, an essay on the application of mathematical analysis to the theories of electricity and magnetism. Published at Nottingham,

in

1828. Mathematical Papers of George Green, Chelsea Pub. Co., New York, (1970)

.

/4/ Works of Benjamin Franklin, edited by J.

Sparks,

Boston (1837), Vo1.V.

/5/ The Feynman Lectures on Physics Vol.11. A&lison-Wesley Pub. Co. (1964)

.

/6/ J.C.Maxwel1, A Treatise on Electricity and Magnetism, thinl ed. 1891,

Dover

Pub. New York, (1954) 152-154.

/7/ M-Faraday, Experirrental Researches i n Electricity, (1838) Londan.