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- Consider two thin current loops. Let loops
and
carry the currents
and
, respectively. The
magnetic force exerted on loop 2 by loop 1 is [see Equation (616)]

where
. Here,
and
are the position vectors of elements of loops
and
,
respectively. Demonstrate that the previous expression can also be written

Hence, deduce that

in accordance with Newton's third law of motion.

- Consider the two current loops discussed in the previous question. The magnetic field generated at a general position vector
by the
current flowing around loop
is [see Equation (614)]

Demonstrate that

where

Show that the magnetic flux passing through loop
, as a consequence of the current flowing around loop
, is

Hence, deduce that the mutual inductance of the two current loops takes the form

- The vector potential of a magnetic dipole of moment
is given by

Show that the corresponding magnetic field is

- Demonstrate that the torque acting on a magnetic dipole of moment
placed in a uniform
external magnetic field
is

Hence, deduce that the potential energy of the magnetic dipole is

- Consider two magnetic dipoles,
and
. Suppose
that
is fixed, whereas
can rotate freely in any direction. Demonstrate that the equilibrium configuration of the second dipole is
such that

where
and
are the angles subtended by
and
, respectively, with the radius vector joining them.

** Next:** Magnetostatics in Magnetic Media
** Up:** Magnetostatic Fields
** Previous:** Localized Current Distribution
Richard Fitzpatrick
2014-06-27