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# Exercises

1. 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.

2. 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

3. The vector potential of a magnetic dipole of moment is given by

Show that the corresponding magnetic field is

4. 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

5. 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