Mathematics of Physics and Engineering

168 Maxwell's Equations

By (3.3.2), div.E = p/e 0. The first equality in (3.3.14) then implies

V^2 t/ = - —, (3.3.15)
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that is, the potential of the electric field satisfies the Poisson equation
(3.3.15) in K^3. Our arguments also suggest that the second equality (3.3.14)
gives a solution of (3.3.15); see Problem 4.2 on page 427 for the precise
statement and proof of the corresponding result.
EXERCISE 3.3.7. (a)c Verify that (3.3.15) follows from (3.3.2). (b)B
Taking for granted that the function U defined in (3.3.14) is twice con-
tinuously differentiable in M.^3 , use (3.3.15) to show that U is a harmonic
function outside of some bounded domain. Hint: recall that we assume that
p = 0 outside of some bounded domain.
The analogs of (3.3.2), (3.3.14), and (3.3.15) exist for every central,
inverse-square field, such as the gravitational field.
EXERCISE 3.3.8.c Let E = E{P), P e M^3 , be the gravitational field
intensity vector, so that F = mE(P) is the force acting on a point
mass m placed at the point P. Verify that

E = -VV and V^2 V = 4TTGP, (3.3.16)

where G is the universal gravitational constant (see page 45) and p is the
density, per unit volume, of the mass producing the gravitational field. Hint:
for a point mass M at the origin, E{P) = —MGr~^2 r = MGV(l/r). Then re-
peat the arguments that lead from (3.3.7) to (3.3.9) and from (3.3.13) to (3.3.15).
Let us now consider the magnetic field. By Maxwell's equation (3.3.4),
the vector field B is solenoidal, that is, has zero divergence. The condition
div B — 0 is satisfied if

B = cm\A (3.3.17)

for some vector field A; if we can find A, then we can find B. This field

A is called a vector potential of B. Note that, if it exists, A is not

unique: for every sufficiently smooth scalar field /, A\ = A + grad / is

also a vector potential of B, because curl(grad/) = 0. On the other hand,

div A\ = div A + V^2 /; with a suitable choice of / we can ensure that the

vector potential satisfies

div^4 = 0. (3.3.18)