Mathematics of Physics and Engineering

Equations in Material Media 173

Denote by n the unit vector in the direction of OQ x u, and by a, the
radius of the circle.

EXERCISE 3.3.16.B (a) Verify that the direction of n is the same for every
point Q of the circle. How should a small bar magnet be placed to produce
the same magnetic field as the magnetic dipole? Hint: draw a picture; ifC is
a circle in the (i, j) plane and is oriented counterclockwise, then n = k. (b)
Verify that (3.3.31) can be written as

Hint: to show that fcuds = 0, use cartesian coordinates so that C is a circle in
the (i, j) plane and is oriented counterclockwise. Then u = — sinti + cost j and
ds = adt, 0 < t < 2-7T. For the second integral, use (3.2.14) on page 157 with
ro = OP, r = OQ, and S, the disk enclosed by C.

Similar to the electric dipole, we define the dipole moment of the mag-
netic dipole as the vector m = -Ka^2 In. In the limit a —> 0 and I —> oo
so that the vector m stays the same, we get the point magnetic dipole.
If a point magnetic dipole is placed at the origin, then the corresponding
vector potential is

EXERCISE 3.3.17? Find the magnetic field B of the point magnetic dipole.

3.3.3 Maxwell's Equations in Material Media

In conductors, such as metals and electrolytes, there are charged particles
that can move when an external electric field E is applied. In metals, these
particles are free electrons, and in electrolytes, ions. By Ohm's Law, the
resulting current density J is proportional to the external field E:

J = aE, (3.3.34)

where a is the conductivity of the material. Similar to other laws of elec-
tricity and magnetism, relation (3.3.34) is a mathematical expression of
empirical observations and was originally published in 1827 by the German
physicist GEORG SIMON OHM (1789-1854). Today, (3.3.34) is known as
the microscopic form of Ohm's Law, as opposite to the more familiar macro-