Page 150 - Electromagnetics

Figure 3.9: Green’s function for a grounded conducting sphere.

Note that 2 → 1 as 2 → 1 .
There is an image interpretation for the secondary Green’s functions. The secondary
Green’s function for region 1 appears as a potential produced by an image of the primary
charge located at −z in an infinite medium of permittivity 1 , and with an amplitude of

( 1 − 2 )/( 1 + 2 ) times the primary charge. The Green’s function in region 2 is produced
by an image charge located at z (i.e., at the location of the primary charge)in an infinite

medium of permittivity 2 with an amplitude of 2 2 /( 1 + 2 ) times the primary charge.

Example solution to Poisson’s equation: conducting sphere. As an example
involving a nonplanar geometry, consider the potential produced by a source near a
grounded conducting sphere in free space (Figure 3.9). Based on our experience with
planar layered media, we hypothesize that the secondary potential will be produced by
an image charge; hence we try the simple Green’s function

A(r )
s
G (r|r ) =

4π|r − r |
i
where the amplitude A and location r of the image are to be determined. We further

i
assume, based on our experience with planar problems, that the image charge will reside
inside the sphere along a line joining the origin to the primary charge. Since r = aˆ r for
all points on the sphere, the total Green’s function must obey the Dirichlet condition

1 A(r ) 1 A(r )

G(r|r )| r=a = + = + = 0

4π|r − r | r=a 4π|r − r | r=a 4π|aˆ r − r ˆ r | 4π|aˆ r − r ˆ r |

i
i
in order to have the potential, given by (3.56), vanish on the sphere surface. Factoring a
from the first denominator and r from the second we obtain

i

1 A(r )
+ = 0.
r a
ˆ

4πa|ˆ r − r | 4πr | ˆ r − ˆ r |
a i r i
2
2

Now |kˆ r − k ˆ r |= k + k − 2kk cos γ where γ is the angle between ˆ r and ˆ r and k, k

are constants; this means that |kˆ r − ˆ r |=|ˆ r − kˆ r |. Hence as long as we choose

r a A 1
= , =− ,
a r r a
i i
the total Green’s function vanishes everywhere on the surface of the sphere. The image
charge is therefore located within the sphere at r = a r /r 2 and has amplitude A =
2

i
© 2001 by CRC Press LLC