Mathematics of Physics and Engineering

166 Maxwell's Equations

a detailed explantation and generalizations of this phenomenon. In 1827,
he gave a mathematical formulation, connecting the steady electric current
with the induced magnetic field. In modern terms, the formulation is as
follows Let 5 be a piece-wise smooth orientable surface with a piece-wise
smooth boundary OS so that the orientations of S and dS agree. If J is the
density, per unit area, of the stationary (that is, time-independent) electric
current, then the induced magnetic field B satisfies £„„ B-dr = fio JJ J-dcr;
s
see also Exercise 3.3.11 on page 170 below. By Stokes's Theorem, this
implies Ampere' s Law

curlB = /xoJ; (3.3.11)

Maxwell used the equation of continuity (3.2.8) and his first equation (3.3.2)
to extend (3.3.11) to time-varying currents in the form (3.3.5). Recall that
div (curl F) = 0 for every twice continuously differentiable vector field F.
Then, assuming B is twice continuously differentiable, we get from (3.3.11)
that div J = 0. On the other hand, if p is the density, per unit volume, of
the charges moving with velocity v, then J = pv, and, by the equation of
continuity with no sources and sinks,

^+divJ = 0, (3.3.12)

or div J = —dp/dt. For a stationary current, dp/dt = 0, which is consistent

with div(curlJB) = 0. For time-varying currents, we use (3.3.2) to find

dp/dt = e 0 div(dE/dt) so that div (J + e 0 dE/dt) = 0. To ensure the

equality div(curlB) = 0, we therefore replace (3.3.11) with (3.3.5).

EXERCISE 3.3.3.B GO over the above arguments and verify that the diver-

gence of the right-hand side of (3.3.5) is indeed equal to zero.

Equations (3.3.4) and (3.3.5) express the observed interdependence be-

tween B and E, leading to the single theory of electromagnetism. This

is a striking example of the power of mathematical modelling in describ-

ing physical phenomena. A successful model can also predict new physical

phenomena, and we will see later how Maxwell's equations lead to the pre-

diction of electromagnetic waves (see page 348 below).

We conclude this section by establishing the connection between a sta-

tionary electromagnetic field and the Poisson equation. We start with the

stationary electric field. Recall that grad(l/r) — —r/r^3. Equation (3.3.7)

therefore implies that the electric field E produced by a single point charge