Page 137 - Electromagnetics

normal component of D on a conductor is equivalent to specification of the surface
charge density. Thus we must specify the potential or surface charge density over all
conducting surfaces.
One other condition results in zero on the left-hand side of (3.44). If S recedes to
infinity and 0 and D 0 decrease sufficiently fast, then (3.45)still holds and uniqueness
2
is guaranteed. If D, E ∼ 1/r as r →∞, then ∼ 1/r and the surface integral in (3.44)
2
tends to zero since the area of an expanding sphere increases only as r . We shall find
later in this section that for sources of finite extent the fields do indeed vary inversely
with distance squared from the source, hence we may allow S to expand and encompass
all space.
For the case in which conducting bodies are immersed in an infinite homogeneous
medium and the static fields must be determined throughout all space, a multiply-
connected surface is used with one part receding to infinity and the remaining parts
surrounding the conductors. Here uniqueness is guaranteed by specifying the potentials
or charges on the surfaces of the conducting bodies.

3.2.4 Poisson’s and Laplace’s equations

For computational purposes it is often convenient to deal with the differential versions

∇× E(r) = 0, (3.46)
∇· D(r) = ρ(r), (3.47)

of the electrostatic field equations. We must supplement these with constitutive relations
between E and D; at this point we focus our attention on linear, isotropic materials for
which
D(r) = (r)E(r).

Using this in (3.47)along with E =−∇ (justified by (3.46)), we can write
∇· [ (r)∇ (r)] =−ρ(r). (3.48)
This is Poisson’s equation. The corresponding homogeneous equation

∇· [ (r)∇ (r)] = 0, (3.49)

holding at points r where ρ(r) = 0,is Laplace’s equation. Equations (3.48)and (3.49)
are valid for inhomogeneous media. By (B.42)we can write

∇ (r) ·∇ (r) + (r)∇· [∇ (r)] =−ρ(r).
2
For a homogeneous medium, ∇ = 0; since ∇· (∇ ) ≡∇ ,wehave
2
∇ (r) =−ρ(r)/ (3.50)
in such a medium. Correspondingly,
2
∇ (r) = 0
at points where ρ(r) = 0.
Poisson’s and Laplace’s equations can be solved by separation of variables, Fourier
transformation, conformal mapping, and numerical techniques such as the finite difference
and moment methods. In Appendix A we consider the separation of variables solution

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