Exercises

1. Consider two thin current loops. Let loops $1$ and $2$ carry the currents $I_1$ and $I_2$ , respectively. The magnetic force exerted on loop 2 by loop 1 is [see Equation (616)]
   $${\bf F}_{21} = \frac{\mu_0\,I_1\,I_2}{4\pi}\oint_1\oint_2\frac{d{\bf r}_2\times (d{\bf r}_1\times {\bf r}_{12})}{\vert{\bf r}_{12}\vert^{\,3}},$$
   where ${\bf r}_{12} = {\bf r}_2-{\bf r}_1$ . Here, ${\bf r}_1$ and ${\bf r}_2$ are the position vectors of elements of loops $1$ and $2$ , respectively. Demonstrate that the previous expression can also be written
   $${\bf F}_{21} =- \frac{\mu_0\,I_1\,I_2}{4\pi}\oint_1\oint_2\frac{(d{\bf r}_1\cdot d{\bf r}_2)\,{\bf r}_{12}}{\vert{\bf r}_{12}\vert^{\,3}}.$$
   Hence, deduce that
   $${\bf F}_{12} = -{\bf F}_{21},$$
   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 ${\bf r}$ by the current flowing around loop $1$ is [see Equation (614)]
   $${\bf B}({\bf r}) = \frac{\mu_0\,I_1}{4\pi}\oint_1 \frac{d{\bf r}_1\times ({\bf r}-{\bf r}_1)}{\vert{\bf r}-{\bf r}_1\vert^{\,3}}.$$
   Demonstrate that
   $${\bf B} = \nabla\times {\bf A},$$
   where
   $${\bf A}({\bf r})= \frac{\mu_0\,I_1}{4\pi}\oint_1\frac{d{\bf r}_1}{\vert{\bf r}-{\bf r}_1\vert}.$$
   Show that the magnetic flux passing through loop $2$ , as a consequence of the current flowing around loop $1$ , is
   $${\mit\Phi}_{21} = \frac{\mu_0\,I_1}{4\pi}\oint_1\oint_2 \frac{d{\bf r}_2\cdot d{\bf r}_1}{\vert{\bf r}_1-{\bf r}_2\vert}.$$
   Hence, deduce that the mutual inductance of the two current loops takes the form
   $$M = \frac{\mu_0}{4\pi}\oint_1\oint_2 \frac{d{\bf r}_1\cdot d{\bf r}_2}{\vert{\bf r}_2-{\bf r}_1\vert}.$$

3. The vector potential of a magnetic dipole of moment ${\bf m}$ is given by
   $${\bf A}({\bf r}) = \frac{\mu_0}{4\pi}\,\frac{{\bf m}\times {\bf r}}{r^{\,3}}.$$
   Show that the corresponding magnetic field is
   $${\bf B}({\bf r}) = \frac{\mu_0}{4\pi}\left[\frac{3\,({\bf r}\cdot{\bf m})\,{\bf r}- r^2\,{\bf m}}{r^{\,5}}\right].$$

4. Demonstrate that the torque acting on a magnetic dipole of moment ${\bf m}$ placed in a uniform external magnetic field ${\bf B}$ is
      $\tau$ $$= {\bf m}\times {\bf B}.$$
   Hence, deduce that the potential energy of the magnetic dipole is
   $$W = - {\bf m}\cdot{\bf B}.$$

5. Consider two magnetic dipoles, ${\bf m}_1$ and ${\bf m}_2$ . Suppose that ${\bf m}_1$ is fixed, whereas ${\bf m}_2$ can rotate freely in any direction. Demonstrate that the equilibrium configuration of the second dipole is such that
   $$\tan\theta_1 = - 2\,\tan\theta_2,$$
   where $\theta_1$ and $\theta_2$ are the angles subtended by ${\bf m}_1$ and ${\bf m}_2$ , respectively, with the radius vector joining them.