Mathematics of Physics and Engineering

170 Maxwell's Equations

where u is the unit vector along the wire in the direction of the current,

Q is the point on the wire closest to P, and J is the current measured in

amperes. In particular, the magnitude of B is inversely proportional to the

distance from the wire.

EXERCISE 3.3.11. (a)B Draw a picture illustrating (3.3.22) and derive

(3.3.22) from (3.3.21). Hint: consider a uniform cylindrical wire of small ra-

dius. (b)c Let C be a circle of radius R in the plane perpendicular to the

wire and center the wire. Show that (3.3.22) implies § B • dr = fi 0 I- (c)B

C

Show that the result of part (b) is true for every simple closed piece-wise

smooth curve enclosing the wire.

EXERCISE 3.3.12^ LetC be a simple piece-wise smooth curve representing an

infinitesimally thin conducting wire with current I (measured in amperes);

the value of the current does not depend on time but can be different at

different points of the wire. Assume that orientation ofC is in the direction

of the current. Use (3.3.21) to show that the resulting magnetic field B and

the corresponding vector potential A at a point P not in C are

where Q is a (varying) point on C and u is the unit tangent vector to C.

Hint: assume that the wire has a small radius a and is bent as C. Then ira^2 J = Iu

andfff(---)dV = f(-.-)Tra^2 ds.

3.3.2 The Electric and Magnetic Dipoles

As an example illustrating some of the general discussions from the previous
section, we will consider electric and magnetic dipoles. In the following
section, these dipoles will help us to study the electromagnetic field in
material media.
An electric dipole is a pair of point charges q and — q, q > 0, with
equal magnitude and opposite signs, placed at a fixed distance from each
other. Let O be a reference point in R^3 , and let the position vectors of the
charges q, -q be r\ and r 2 , respectively. By (3.3.13), the potential U of
the resulting electric field at the point with position vector r is

U(r) = -?- ( l -^1 ). (3.3.24)

47TE0 Vlk-^lll \\r-rA)

Let ro = {r\ + T2)/2 be the position vector of the midpoint of the line