JOURNAL OF APPLIED
PHYSICS
VOLUME
21.
NUMBER
8
AUGUST.
1956
Charged Right Circular Cylinder
W. R. SMYTHE
Norman Bridge Laboratory, California Institute of Technology, Pasadena, California
(Received March 5, 1956)
A new method permits the calculation of the electric field surrounding a charged conducting surface of
revolution without the use of orthogonal functions. Detailed formulas show how to find the charge density
on a right circular cylinder with any desired precision. The numerical examples worked out give the maximum deviation of the actual surface from that of a true cylinder of diameter d to be -0.OO15d, -0.OOO37d,
-0.OO017d, +0.OO16d, and +0.014d for length to diameter ratios 1, I, 1, 2, and 4, respectively. The capacitance calculation gives an accuracy of one part in 30000 for the ratios I, 1, and 2. A capacitance formula is
worked out which is accurate to one part in 1000 over the ratio range 0 to 4. Additional formulas indicate
the method of solution for the bodies in longitudinal and transverse electric fields and the extension of twobody problems such as the thick plate parallel plate capacitor. A way to calculate the flow about bodies of
revolution is indicated. Digital computers are well suited to this method as no function tables are needed.
1. INTRODUCTION
of the most common figures occuring in apO NEparatus
is the right circular cylinder. Thus, its
capacitance and the fields surrounding it when it is
charged are of considerable practical importance. The
reason the literature is blank on this subject is evident
when one sees the hopeless complications the usual
orthogonal function techniques involve when applied
to this problem. A new approach is needed. The method
to be described gives a complete solution, as accurate as
desired, of this and similar problems.
The charge density CT. on the end must be a function of
p2 to have a minimum (dCT./dp=O) at the pole and that
on the side (CT.) must be a function of Z2 to have a
minimum (dCT./dz=O) at the equator. Furthermore,
very near the edge the charge densities become equal
and approach that on the charged rectangular wedge
which is proportional to o-t where ~ is the distance from
the edge. Suitable forms are therefore
co
CT.=L: An(b2-z12)n- i
(4)
n=O
co
2. METHOD OF SOLUTION
CT.=L: Bn(1-PI2)n-i,
An axially symmetrical solution of Laplace's equation in the cylindrical coordinate system P, 11', z is
(5)
n=O
where b, ZI, and PI are dimensionless numbers given by
(6)
(1)
Differentiation 2p times with respect to z introduces a
factor (lP into the integrand. Thus, when the Bessel
function is expanded in the usual way in even powers
of tp, the result may be written
This expression is widely used in electrostatic electron
microscope lens calculations and gives, in terms of the
potential V(z,O) on the axis, the potential at any point
off the axis which can be reached without crossing a
charge sheet. The charge distribution on a closed conducting surface that encloses the origin gives a constant
potential Vo inside so that
P
d2 V(Z,O)]
[
dz2
p
.-0
=
{VO when p=O
(3)
0 when p,.r:O.
The important n=O terms will be called the "fundamental" terms and the remaining ones the "correction"
terms. The latter are zero at the edge and in order that
the fundamental terms match there it is necessary that
Ao=blBo.
(7)
The number of correction terms required for a given
accuracy is much smaller for the forms given in Eqs. (4)
and (5) than when integral powers of p and z are used.
Also, as will appear shortly, numerical checking is
simpler. When the potential on the axis due to the
charge densities of Eqs. (4) and (5) is calculated,
differentiated with respect to ZI and ZI is equated to
zero the result is
d2 p V 00 MBn
-=L: --F[p+t, n-p+i; n+1+i; (1+b2)-IJ
dZ 12p n=O n+i
+MNGA nF[P+t, n- p+i; n+ 1+i; b2(1 +b2)-IJ, (8)
where
Thus our problem is to find a charge distribution on the
surface of the right circular cylinder bounded by the
planes z=±c and the cylinder p=a which satisfies (3).
b2n+i(n-i) !(p-t)!
ta(2p)!
M
E(l +b2)P+t'
N
(-1)p2 f (n+i) !P!
917
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W.
918
R.
and
G= [r (l)J2[rmjl =0.684463408,
where F(a,b j C j z) is a hypergeometric function. For
simplicity, negative factorials are taken unity throughout this paper so that (n-i)! means 1·i· 5/3· .. (n-i).
The units are mks so a is in meters, V in volts, and E is
8.85434X 10-12 f/m.
SMYTHE
viation. From symmetry one of these (of zero radius)
is at the pole and one at the equator and the deviations
at these points should have the same sign when p is
odd and opposite signs when p is even. From Eq. (2)
the amplitude of these deviations should increase with
increasing distance from the origin. Except at the
TABLE
3. DETERMINATION OF COEFFICIENTS
b=l
The determination of Ao( =b1Bo), AI, .. " Ar and B 1,
B 2, •• " B. in Eqs. (4) and (5) requires writing the
equations in (3) for p values 0, 1, 2, .. ·r+s and solving
these equations simultaneously. The chief labor is the
calculation of the hypergeometric functions F(a,b j C j z).
This is greatly facilitated by use of the following recursion formulas.
c(c-1)F(a, b-1 j c-1 j z)+c[ (a-b)z-c+ 1J
XF(a,b j C j z)+b(c- a)zF(a, b+ 1 j c+ 1 j z) =0
cF(a,bj Cj z)-cF(a+1, b-1 j Cj z)
- (a-b+1)zF(a+1,
c(a-b)F(a,bj
Cj
(9)
A 0=0.39326697
A 1 = 10.911218
A2= -181.16229
Bo=0.62427242
B 1 = -0.09128092
B. = 0.02058206
EV
Potentials
EV Pole
1.00000002 V
EV Equa. 0.9923 V
EV Edge 1.00079692 V
EV
Capacitance
10.420264 Ea
EV
A o=0.40304719
Al = 6.4134459
A 2 = -97.322125
B o=0.63979755
B 1 = -0.13360489
B. = 0.06215840
B3= -0.01398595
Potentials
EV
EV Pole
1.00000000 V
EV Equa. 0.9968 V
Edge
0.99972774
V
EV
EV
Capacitance
EV
10.4155203 Ea
.V
A 0=0.46928422
Al =0.89919472
A.= -2.4818068
B o=0.59126109
B 1 = -0.09703091
B.=0.02927739
.V
Potentials
.V Pole
1.00000395 V
EV Equa. 0.99937427 V
.V Edge 0.99725336 V
.V
Capacitance
EV
12.107972 .a
A 0= 0.55905946
A 1=0.14867157
A.= -0.05617693
Bo = 0.55905946
B 1 = -0.14082266
B 2 =0.07155254
EV
Potentials
EV Pole
1.00010243 V
EV Equa. 0.99997755 V
EV
EV
Capacitance
.V
14.973964 fa
A 0=0.67570982
A 1=0.03921759
A.= -0.022289015
Bo=0.53631121
B 1 = -0.38793856
B 2 =0.35626582
.V
Potentials
.V Pole
1.00440238 V
.V Equa. 0.99999954 V
EV
.V
Capacitance
.V
19.776506 Ea
A 0=0.66050233
A 1=0.05455850
A 2 = -0.027136699
A 3 = 0.024926044
B o=0.52424102
B 1 = -0.23638748
B.=0.18391OO1
EV
Potentials
EV Pole
1.00149239 V
EV Equa. 1.00000001 V
EV
EV
Capacitance
EV
19.775418 Ea
.V
A o=0.83974760
Al =0.012420258
A.= -0.021371704
B o=0.52900782
B 1 = -2.1617990
B.=2.8472879
EV
Potentials
.V Pole
1.0893612 V
• V Equa. 0.99999935 V
EV
.V
Capacitance
EV
27.745204 Ea
A o=0.79451208
A 1=0.020672127
A.= -0.()374287843
A 3 =0.()419885966
A.=-0.0623777893
B o=0.50051123
B 1 = -0.65430090
B 2 =0.73249240
.V
EV
EV
EV
EV
EV
EV
EV
Deviations
+O.OOOOOOa
-0.0083a
+0.OOOO25a
Deviations
+O.OOOOOOa
-0.003Oa
-0.000005a
b=i
bj
c+1 j z)=O
z)-a(c-b)F(a+1, bj c+1; z)
+b(c-a)F(a, b+1 j c+1 j z)=O.
(10)
(11)
Only four functions need be calculated by series, say
p=O for AoAl, B o, B 1• The remainder are then found
by repeated application of Eqs. (10) and (11). The
maximum p value (r+s) terms are checked by Eq. (9).
If these check to eight places, all intermediate p values
are good to eight places and the calculation need not
be repeated. If these fail to check the error can be
located by application of Eq. (9) to intermediate
values. These operations are simple and fast on a
digital computer. Little labor is involved in checking
the factors that multiply F(a,bj Cj z) in Eq. (8) by
repetition of the calculation. The first function in Eq.
(8) converges fast for large values of b and the second
for small values. For hand calculation of other values
analytic continuation formulas 1 save time. Sets of coefficients for b values t, !, 1, 2,4 are given in Table I.
One must use the complete set for omission of any term
may cause large errors. Sets of from four to six give
adequate accuracy over this range. Eight places were
carried in all calculations so the last digit is uncertain
due to cumulative rounding off errors.
4. ACCURACY OF RESULTS
To evaluate the results some estimate of accuracy is
needed. If there are m-correction terms then there are
m+ 1 constants which are adjusted to some optimum
value by Eq. (3). The actual Vo surface can intersect
the cylinder on only m+ 1 circles between pole and
equator. These separate m+ 2 circles of maximum deErdelyi, Magnus, Oberhettinger, and Tricomi, Higher Transcendental Functi9ns (McGraw-Hill Book Company, Inc., New
York, 1953), Vol. 1, Eq. (1) p. 108, and Eq. (12) p. 110.
1
I."
Deviations
+0.0000075a
-0.000732a
-0.000100
b=l
Deviations
+0.000209a
-0.OOOO344a
b=2
Deviations
+0.00872a
-0.000001 a
Deviations
+0.00316a
-0.000000a
b=4
Potentials
Pole 1.01596239 V
Equa. 0.99999942 V
Deviations
+0.1496a
-0.000002a
Deviations
+0.02700
-0.000001 a
Capacitance
27.844690
fa
• Note that 0.0'3 means 0.0003.
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CHARGED
RIGHT CIRCULAR
edge, the distance between the actual surface and the
cylinder is found by multiplication of the potential
difference at that point by eju. These deviations are
shown in Table I. It is simpler to calculate the coefficients than the potentials. The potential at the pole
due to the A .. side term is
alrn+l(n-t) !GA"
00
(s-!) !lr'
2E21(1 +lr)l
s=O
(n+s+t) !(1 +b )8
CYLINDER
wedge existed there which is strictly true only for
positive deviations.
5. POTENTIAL OF CHARGED CYLINDER
The potential outside the charged cylinder is
V(p,\O,z)=
-----L -----2
XP2,[b(1 +lr)-lJ,
(12)
where P 2 ,(x) is a Legendre polynomial. For hand calculation for large b values a much more complicated but
rapidly convergent formula was used. The pole potential
due to the Bn end terms is given by
aBn{ (n-t)!G +FCt,n+ i ;n+(5/3); (1+4b2)-IJ}.
(n+i) (1 +4b2)1
4E 2f(n+t)!
ab2n+lGA ..
00
21r21E
s=O
X [In(b-18) +!¢t(s+n+ ll)-!¢t(s+!)-N ,J
n
(14)
where
No=O,
1
N,=2 -1
+-+
... +1 ].
[ 1·2 3·4
2s(2s-1)
For large values of b this was replaced by
ab2n+l(n-t) !GAn
00
2E(1 +b )t2t(n+t)!
,91
[(s-!) !J2
- - -2 - - - L - - - 2
(s !)2(1 +b
)'
XF[s+!, n-s+t; n+l+t; b2(1+b 2)-I].
(15)
The potential at the equator due to the two ends is
(16)
This converges slowly for small b values for which no
formula suitable for hand calculation to eight places
was found. Note that only four places are given for the
equatorial potential when b=i.
The potential at the edge is of no value in error
estimation as, except by accident, it falls on neither a
maximum nor a minimum. Its value is of some interest,
however, and is given for small b values where a good
formula was available. The actual distance between the
Vo surface and the cylinder at the edge is small because
of the large value of u there. The edge deviation given
in the table is calculated as if the field of a right angle
(17)
The capacitance is found by integration of Eqs. (4)
and (5) over the surface to get the total charge and
division by V o which give
C=1raE L [
(s!)2(s+n+t)W
f
6. CAPACITANCE
[(s-!)!J3(n-t)!(-b 2)'
--- L --------
udS
--,
s 41rEr
where r is the distance from the field point to the surface element dS and u is u. on the ends (5) and u, on
the sides (4). An expression for the potential of the
ends is easily found as a sum of spherical harmonics by
taking the origin for each end at its center and integrating over ring elements. For the sides, however, all
integrals found were double sums and none was valid
over the whole external region. It is probable one must
resort to numerical integration.
(13)
For small values of b an analytic continuation formulal was used. For small values of b the potential at
the equator due to the sides is
919
2Bn
(2b)ib 2n (n-t) !GAn]
n+t
(n+t)!
--+
.
(18)
The error in this value should be much smaller than the
surface deviations because positive and negative deviations tend to cancel. Thus at b= t a change in the equatorial potential deviation from 0.77% to 0.32% produces a change in C of only 0.0047%. Possibly even and
odd numbers of correction terms give different sign
errors in the capacitance because of a reversal in the
dominance of positive and negative deviations. Note
that successive correction terms alternate in sign. In
any case more terms give greater accuracy. The following capacitance values are believed to be significant in
the last digit. The thin disk for which the capacitance
is exactly 8Ea is inserted for completeness. We write
cia for b.
cia
o
t
t
1
2
4
C in farads
0.708347 X 10-10a
0.9222 X 10-loa
1.07208 X 10-10a
1.32585 X 10-10a
1.75098 XIo- l oa
2.465
X 10-10a.
It would be useful to have some capacitance formula
valid for intermediate values in this range. An approximate capacitance formula may be found from Eqs. (3),
(4), (5), and (7) using only the fundamental terms. It
gives values which deviate smoothly from the more
exact values and so can be corrected by a simple addi-
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w.
920
R.
tive term. Thus, we obtain
_
1
C=af: - - - - - - - - - - - - - - - -
F(!,
Ii l+Ii x)+1.7248b i F(!, Ii l+*i Y)
+0.714 exp( -1.55b l )-0.232b4 / 3 ] farads,
(19)
where x is (1+b2)-t, y is b2(1+b2)-1, b is the ratio of
length to diameter and a is the radius in meters. The
accuracy is one part in 1000 or better over the range
0<: b<: 4. In the ranges of x and y where the hypergeometric series converge slowly analytic continuation
formulas1 will give rapid convergence.
7. OTHER APPLICATIONS
Evidently the foregoing calculations can be carried
out for other charged solids of revolution provided
suitable expansions such as (4) and (5) can be found
for the surface charge density. It is by no means certain
that those used are best, even for a cylinder. Taylor
has suggested certain orthogonal polynomials which
may be better. The problem of the solid in an axial
uniform field is easily solved by differentiation of Eq.
(2) with respect to z so that
-Ez= aV(z,p)
az
=I: (-1)P(~)2Pd2P.HV(Z'0).
0
(p 1)2
00
V(p,cp,z)-coscp L
12.566(1 +b2)l(1 + 1. 7248b l )
[
SMYTHE
2
dZ
(20)
2 P+l
Instead of Eq. (3) one now requires that E. cancel the
applied field inside the cylinder and that higher derivatives vanish. In the cylinder case, this leads to almost
identical hypergeometric functions but odd instead of
even z dependence. For a traverse field one differentiates Eqs. (1) and (2) with respect to p, integrates with
respect to z, and multiplies by coscp so that
(p)2P-l d2p-l V (z,O)
.
pl(p-l)1 2
dZ 2p-l
(-l)p
(21)
Instead of Eq. (3) one now requires that V=Ep cOScp
cancel the applied potential of the uniform field Ell:
inside and that higher derivatives vanish. Potential
flow around a body may be found in exactly the same
way by using the vector potential expansion2 analogous
to Eq. (2) and requiring that A", be zero inside the
body which makes the boundary a line of flow.
The same method applies to two identical mutually
external coaxial bodies of rotation with equal charges
of the same or opposite sign. For example, the parallel
plate capacitor problem where each plate is a short
cylinder can be solved by Eq. (3) by taking the origin
at the center of one of the cylinders and adding to
d2 p V jdz2p the contribution from the image charge distribution on the other cylinder. Two equations like
Eq. (5) will be needed because Bn will be different on
the two cylinder faces. Lack of symmetry on the sides
will require two factors (b-z1)n-l and (b+z1)n-l instead of the single factor W-zl)n-l in Eq. (5). Thus
for the same accuracy twice as many coefficients will
probably be needed as for the single cylinder. There is
also another parameter, the distance between cylinders.
Digital computers make such calculations feasible.
There have been experimental indications that the old
Kirchoff formula 3 is badly off in some cases. Other
problems such as a charged right circular cylinder
inside a coaxial conducting tube, potential flow about a
right circular cylinder inside a coaxial pipe, etc., can
be set up in similar fashion. Useful discussions on this
problem have been held with Taylor who is working on
the application of a similar method to electromagnetic
wave antenna problems.
I W. R. Smythe, Static and Dynamic Electricity (McGraw-Hill
Book Company, Inc., New York, 1950), Eq. (5), p. 268.
8 H. Geiger and Karl Scheel, Handbuch der Physik (Verlag
Julius Springer, Berlin, 1927), Vol. XII, p. 485.
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