MASSACHUSETTS INSTITUTE OF TECHNOLOGY
Department of Physics: 8.02
Capacitance of Concentric Spherical Conductors
A spherical conductor of radius a carries a charge +Q . A second thin conducting spherical shell
of radius b carries a charge !Q . Calculate the capacitance .
Solution:
The shells have spherical symmetry so we need to use spherical Gaussian surfaces. Space is
divided into three regions (I) outside r ! b , (II) in between a < r < b and !
(III) inside r ! a . In each region the electric field is purely radial (that is E = Er̂ ).
Region I: Outside r ! b :
Region III: Inside r ! a :
These Gaussian surfaces contain a total charge of 0, so the electric fields in these regions must be
0 as well.
Region II: In between a < r < b : Choose a Gaussian sphere of radius r . The electric flux on the
! !
2
E
surface is "
"" ! d A = EA = E ! 4# r
The enclosed charge is Qenc = +Q , and the electric field is everywhere perpendicular to the
surface. Thus Gauss’s Law becomes
p. 1 of 2
MASSACHUSETTS INSTITUTE OF TECHNOLOGY
Department of Physics: 8.02
E ! 4" r 2 =
Q
Q
$E=
#0
4"# 0 r 2
That is, the electric field is exactly the same as that for a point charge.
Summarizing:
# Q
! %
r̂ for a < r < b
E = $ 4!" 0 r 2
%0
elsewhere
&
We know the positively charged inner sheet is at a higher potential so we shall calculate
a
! !
!V = V (a) " V (b) = " $ E # d s = " $
a
b
b
a
Q
Q
Q ' 1 1*
dr =
=
"
>0
2
4%& 0 r b 4%& 0 )( a b ,+
4%& 0 r
which is positive as we expect.
We can now calculate the capacitance using the definition
C=
4"# 0
4"# 0 ab
Q
Q
=
=
=
b$ a
!V
Q % 1 1( % 1 1(
$
$
4"# 0 '& a b *) '& a b *)
Note that the units of capacitance are ! 0 times an area ab divided by a length b ! a , exactly the
same units as the formula for a parallel-plate capacitor C = ! 0 A / d . Also note that if the radii b
and a are very close together, the spherical capacitor begins to look very much like two parallel
plates separated by a distance d = b ! a and area
# a + b%
# a + a%
! 4"
= 4"a 2 ! 4"ab
$ 2 &
$ 2 &
2
A ! 4"
2
So, in this limit, the spherical formula is the same at the plate one
4"# 0 ab # 0 4" a 2 # 0 A
.
!
=
b! a b $ a
d
d
C = lim
p. 2 of 2