JOURNAL
OF
APPLIED
VOLUME
PHYSICS
33,
NUMBER
10
OCTOBER
1962
Charged Right Circular Cylinder
W. R.
SMYTHE
California Institute of Technology, Pasadena, California
(Received March 29, 1962)
A paper with the above title appeared in this Journal in 1956 giving the charge distribution on, and
capacitance of, an electrified right circular solid conducting cylinder for length-to-diameter ratios of 0.25
to 4. Much more accurate values, calculated for the range 0.125 to 8 on a digital computer, are tabulated in
the present paper. Over this range the capacitance is given to 0.2% accuracy by the formula
C = [0.708+0.615 (b I a)O.76JX 10-loa farads,
where 2b is the length and a the radius. A spherical harmonic expansion for the potential outside the circumscribed sphere is given.
1. INTRODUCTION
PAPERl with the above title which appeared in
1956 gave the capacitances and charge distributions on solid right circular conducting cylinders with
length-to-diameter ratios ranging from 0.25 to 4. The
labor of solving simultaneous equations and summing
series on a desk computer restricted the results to one
or two coefficient combinations and made error estimates uncertain. A digital computer calculation with
the same formulas, using many coefficient combinations
yielded the improved results recorded here.
A
2. THEORY
The original paperl should be consulted for detailed
formulas. The method used assumes that the charge
densities IT on the sides and IT e on the ends of a cylinder
bounded by z = ± band p = a can be expanded in the
form
$
N
IT.=
11[
L: A n(1-b- 2z2)n-1,
n=O
lTe=
L:
Bm(1-a- 2p2)m-!. (1)
m=O
This An is that of the original paper multiplied by
b2n- 1 so that both An and Bm now have the dimensions
of charge density and are of convenient size. Near the
corners IT, and lTe become infinite properly and match if
(2)
The potential may be expanded at the origin in the form
The potential and its p even derivatives are calculated
from (1). The solution of the simultaneous equations (4)
together with (2) then yield p+2=N+M+2 of the
lowest-order coefficients in (1). The number of signicant digits carried determines the optimum values of
Nand M for a given bla ratio. The best choice is that
for which An and Bm give potentials nearest Vo at pole
and equator. The check point nearest the origin always
gives Vo to six or seven places and is a less sensitive
indicator of the optimum value than the more remote
check point. If M or N is too large, then the contributions of the individual terms, which alternate in sign,
becomes much larger than their sum, greatly reducing
the accuracy of the latter. Most terms in (3) were
found by summing the hypergeometric series and
verified by the recursion formulas.
It should be emphasized that all coefficients must be
used in any field calculation for the omission of any
term may give large errors. There are usually ftom three
to five combinations of A nand Bm in the optimum
range. The one using the fewest coefficients is the one
tabulated and is nearly as accurate as the best combination. For rough values where bla is near one, the results
in the original paper may be used. A calculation of more
than eight place accuracy would be needed to improve
the results in Table r. In many cases the last digit is not
significant because of roundoff errors. The potentials
at pole and equator for the tabulated An and Bm are
given as well as the corresponding proportional displacement I1blb and l1ala of the actual unit potential
surface from that of the cylinder.
00
V(z,p) =
L: (-i p2)P(pl)-2 02PV(Z,0)1 OZ2p
p=o
and it must be constant inside the cylinder so that
where
1
opo
is one if
p=O and zero if po;t.O.
W. R. Smythe, J. Appl. Phys. 27, 917 (1956).
3. FIELD OF CHARGED CYLINDER
(3)
The potential outside the charged cylinder can be
found by integration of ITdSI(47reR) over the surface of
the cylinder, where R is the distance of dS from the
field point. In the region adjacent to the walls it appears
that numerical integration must be used for the side
terms. The potential of the end terms can be expressed as
a series of oblate spheroidal harmonics valid everywhere.
2966
Downloaded 31 Mar 2011 to 134.208.24.193. Redistribution subject to AIP license or copyright; see http://jap.aip.org/about/rights_and_permissions
CHARGED
TABLE
RIGHT
I. Charge density coefficients for unit potential
A 0=- 1.5224294
Bo=0.76121470
B, = -0.38086366
B 2=0.29232287
B3= -0.09010631
CIRCULAR
Outside the circumscribed sphere the potential may be
written as a double series of spherical harmonics.
b=a/8
• V I' =- 1.0000001
.V.=0.9921134
a '"
" (-l)p(C.+C.)(2p+2s)!a 2P b28
v=1: 1 : 2~ s=()
4p(p!)2(2s)! r
28 +2P+l
p=U
tlb =0.OOOOO16b
tla= -0.005a
X P28+ 2p(cos8),
b=a/4
B,= -0.03877087
A 0= 1.0685084
A,=0.1384610
B o=0.67311809
B, = -0.30192902
B 2 =0.41105811
B,= -0.37639902
B.=0.18777757
2967
CYLINDER
where
Iff
.V p= 1.0000001
.V,=l.OOO7242
C.=a
1:
B(p+l, m+l)B""
(6)
m=O
tlb= -0.OOOOOO8b
tla= -0.0006a
N
b=a/2
Ao=0.77698601
A,=0.19084155
A.= -0.16151002
A 3=0.06343914
B o=0.61669421
B, = -0.43141315
B2 = 1.8073652
B3= -5.6072409
B,= 11.453745
B.= -15.478182
B e=13.756435
B 7 = -7.7502333
Bs= 2.5138393
B9= -0.3580848
.V p=0.9999999
• V, = 1.0000026
C,=b
b=a
B3= -3.9209295
B,=6.2012014
B,= -5.6962826
Bs=2.8184609
B 7 = -0.5816914
• V p = 1.0000002
• V. = 1.0000001
tlb= -0.OOOOOO4b
tla= -0.ooOOOO2a
1:
B(s+!, n+l)A n ,
n=O
and B(x,y) is a beta function.
II. Capacitance of right circular cylinder in farads for
various length to diameter ratios, lengths in meters.
TABLE
tlb=O.OOOOOO4b
tla= -0.OOOOO3a
A o=0.55941519
A ,=0.24032463
A 2 = -0.46123818
A.=0.71795706
A,= -0.67534061
A,=0.34357563
As= -0.07271528
B o=0.55941519
Bl = -0.35462716
B2 = 1.4624910
(5)
Capacitance
b/a
0
i
t.l
2
1
2
4
8
0.708347 X 1o0.8312XIo-loa
0.9214X lo-loa
1.07251 X 10-lOa
1.32576X10-10a
1.7 5036X 10-lOa
2.467 X 10-loa
3.7ooXlO-loa
l0 a
Var
Num
Eq. (7)
±O
±8
±3
±6
±2
±3
±1
±6
5
5
3
3
2
4
3
0.708XIo-loa
0.833 X lO-IOa
0.923 X lo-loa
1.072 X lO-IOa
1.323XIo-ioa
1.7 50X lO-IOa
2.472 X 10-loa
3.696X lo-loa
b=2a
A 0=0.40842489
Al =0.23612315
A 2= -0.49824155
A.=1.043774O
A,= -1.6104248
A,=1.7462732
As= -1.2940218
A 7 =0.62430746
As= -0.17688137
A 9 =0.02234862
B o=0.51458312
B 1 = -0.41788413
B 2 = 1.6823388
B3= -3.2845929
B,=2.9374045
B,= -0.9796062
• V p=0.9999970
.V,=0.9999998
tlb=0.OOOOO3b
tla = O.OOOOOO4a
b=4a
A o=0.30232821
A 1=0.19496655
A 2 = -0.20714722
A.=0.17065210
A,= -0.07932248
A 6=0.01551192
Bo = 0.47991610
B 1 = -0.09459217
.V p=0.9954321
.V,=0.9999999
tlb=O.OO4b
tla = 0.OOOOOO3a
b=8a
A o=0.22262099
A 1 =0.20155115
A 2 = -0.17770411
A 3=0.09907634
A.= -0.02279702
B o=0.44524199
.V p= 1.0234341
• V.=1.0000000
tlb= -0.007b
tla= -O.OOOOOOOa
4. CAPACITANCE
At a great distance the potential has the form
so that the charge and hence the capacitance
may be found from the first term (p=O, s=O) of (5) .
This was done for many combinations giving a range of
values wider than that anticipated in the original paper.
The values given in Table II are the means of those
given by all combinations in the optimum range. The
digit in the adjacent column is the amount by which
the last digit must be varied to cover all these combinations including that in Table I. The next column gives
the number averaged. The b=8 capacitance is exactly
8~a. The following very simple formula gives the
capacitance with an accuracy of 0.2% or better over
the range from b=O to b=8a .
q/(47r~r)
. C= [0.708+0.615(b/a)o.76JX 10-10 a farads.
(7)
The values from this formula appear in the last column
of Table II.
Downloaded 31 Mar 2011 to 134.208.24.193. Redistribution subject to AIP license or copyright; see http://jap.aip.org/about/rights_and_permissions