Mathematics of Physics and Engineering

Equations in Vacuum 169

This choice of A corresponds to Coulomb's gauge; for more about gauging
see page 351.
Assume that the current density J and the corresponding electric field
E do not depend on time. Then Maxwell's equation (3.3.5) becomes the
original Ampere's Law (3.3.11).

EXERCISE 3.3.9.B Verify that (3.3.11), (3.3.17), and (3.3.18) imply

V^2 A = -HQJ. (3.3.19)

Hint: see (3.1.32) on page 139.

Let us assume that the vector field J is twice continuously differentiable
and is equal to 0 outside of a bounded domain. By analogy with (3.3.14)
and (3.3.15) (see also Problem 4.2 on page 427), we write the solution of
the vector Poisson equation (3.3.19) as

R^3

with integration over the points QgR^3 where J is not zero.

EXERCISE 3.3.10.c Assuming that differentiation under the integral sign is
justified, verify that (3.3.17) and (3.3.20) imply

R^3
Hint: Note that, for the purpose of differentiating the expression under the integral
in (3.3.20), the point Q is fixed and the point P is variable. Accordingly, place the
origin at the point Q; then you need to compute curl(J/r), where J is a constant
vector. Recall that V(l/r) = —r/r^3 and use a suitable formula from the collection
(3.1.30) on page 139.
Formula (3.3.21) is the three-dimensional version of the Biot-Savart
Law, discovered experimentally in 1820 by the French physicists JEAN BAP-
TISTE BIOT (1774-1862) and FELIX SAVART (1791-1841); in its original
form, the law states that the magnetic field produced by a constant current
/ in an infinitesimally thin straight wire of infinite length is

B{P)-^^PQf' (3- (^3) - 22)