Page 151 - Electromagnetics

−a/r . (Note that both the location and amplitude of the image depend on the location
of the primary charge.)With this Green’s function and (3.71), the potential of an
arbitrary source placed near a grounded conducting sphere is

ρ(r ) 1 1 a/r

(r) = − a 2 dV .

V 4π |r − r | |r − 2 r |
r
The Green’s function may be used to compute the surface charge density induced on
the sphere by a unit point charge: it is merely necessary to find the normal component of
electric field from the gradient of (r). We leave this as an exercise for the reader, who
may then integrate the surface charge and thereby show that the total charge induced
on the sphere is equal to the image charge. So the total charge induced on a grounded

sphere by a point charge q at a point r = r is Q =−qa/r .

It is possible to find the total charge induced on the sphere without finding the image
charge first. This is an application of Green’s reciprocation theorem (§ 3.4.4). According
to (3.211), if we can find the potential V P at a point r produced by the sphere when it is
isolated and carrying a total charge Q 0 , then the total charge Q induced on the grounded
sphere in the vicinity of a point charge q placed at r is given by
Q =−qV P /V 1
where V 1 is the potential of the isolated sphere. We can apply this formula by noting that
2
an isolated sphere carrying charge Q 0 produces a field E(r) = ˆ rQ 0 /4π r . Integration
from a radius r to infinity gives the potential referred to infinity: (r) = Q 0 /4π r. So
the potential of the isolated sphere is V 1 = Q 0 /4π a, while the potential at radius r is

V P = Q 0 /4π r . Substitution gives Q =−qa/r as before.

3.2.5 Force and energy
Maxwell’s stress tensor. The electrostatic version of Maxwell’s stress tensor can be
obtained from (2.288)by setting B = H = 0:
1
¯ ¯
T e = (D · E)I − DE. (3.83)
2
The total electric force on the charges in a region V bounded by the surface S is given
by the relation

¯
F e =− T e · dS = f e dV
S V
where f e = ρE is the electric force volume density.
In particular, suppose that S is adjacent to a solid conducting body embedded in a
dielectric having permittivity (r). Since all the charge is at the surface of the conductor,
¯
the force within V acts directly on the surface. Thus, −T e · ˆ n is the surface force density
(traction) t. Using D = E, and remembering that the fields are normal to the conductor,
we find that
1 1 1
¯ 2 2
T e · ˆ n = E ˆ n − EE · ˆ n =− E ˆ n =− ρ s E.
n
n
2 2 2
The surface force density is perpendicular to the surface.
As a simple but interesting example, consider the force acting on a rigid conducting
sphere of radius a carrying total charge Q in a homogeneous medium. At equilibrium

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