Mathematics of Physics and Engineering

172 Maxwell's Equations

have the potential and the electric field at every point with position vector

r as follows:

U(r) = ~^r ^ E(r) = ^- ( ^-^ - £ ). (3.3.29)

1 dr „,. 1 (Z(d-r)r d

5-, E(r) = ' —

47T£o rd 47T£o

EXERCISE 3.3.15. (a)B Verify that V^2 (d • r/r^3 ) = 0, r ^ 0, and ex-
plain how the result is connected with Maxwell's equation (3.3.2). Hint: use
(3.1.33) on page 140, as well as suitable relations from the collection (3.1.30)
on page 139; note that grad(d • r) = d. (b)A Consider a point electric dipole
with moment d in an external electric field E and assume the electric field
is uniform, that is, has the same magnitude and direction at every point
in space: E(P) = E, where E is a constant vector. Show that the total
force acting on the dipole by the field is zero and the torque is T = d x E.
Thus, the external field will tend to align the dipole with the field. Draw the
picture illustrating the stable orientation of the dipole relative to the field.

MAGNETIC DIPOLE. Since currents produce magnetic fields, we define

a magnetic dipole as an infinitesimally thin closed circular wire in the

shape of a simple smooth closed curve C, carrying a constant current /. By

(3.3.23) on page 170, the vector potential characterizing the magnetic field

of this current is

where u is the unit tangent vector to C in the direction of the current, and
Q is a (varying) point on C. Denote by O the center of the circle C. Then
\PQ\ = \OP — OQ||, and OQ is perpendicular to the tangent vector at the
point Q. As with the electric dipole, we want to get an approximation of
A at the point P that is far away from C, that is, when |OP| is much larger
than the radius \OQ\ of the circle (draw a picture!) Using (3.3.26), we find

1 1 / OP-OQ\

\PQ\ \OP\ \' 1+ \OP\ '^2 J '

Then (3.3.30) becomes

A(P)

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+

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