A sphere of radius R carries charge Q distributed uniformly over the surface, with density σ = Q/4R2. This shell of charge is rotating about an axis of the sphere with angular speed w. Find its magnetic moment. (Divide the sphere into narrow bands of rotat- ing charge; find the current to which each band is equivalent, and its dipole moment, and then integrate over all bands.) (a) We want to find the energy required to bring two dipoles from infinite separation into the configuration shown in Fig. 11.39(a), defined by the distance apart r and the angles 01 and 02. Both dipoles lie in the plane of the paper. Perhaps the simplest way to compute the energy is this: bring the dipoles in from infin- ity while keeping them in the orientation shown in Fig. 11.39(b). This takes no work, for the force on each dipole is zero. Now calculate the work done in rotating m₁ into its final orientation while holding m2 fixed. Then calculate the work required to rotate m2 into its final orientation. Thus show that the total work done, which we may call the potential energy of the system, is equal to (μomim2/4³)(sin₁ sin 02-2 cos 01 cos 02). Spheres of frozen magnetization ** A remarkable permanent magnet alloy of samarium and cobalt has a saturation magnetization of 7.5-105 joule/tesla-m³, which it retains undiminished in external fields up to 1.5 tesla. It provides a good approximation to rigidly frozen magnetization. Consider a sphere of uniformly magnetized samarium-cobalt 1 cm in radius. (a) What is the strength of its magnetic field B just outside the sphere at one of its poles? You can invoke the result from Prob- lem 11.8. (b) What is the strength of its magnetic field B at its magnetic equator? (c) Imagine two such spheres magnetically stuck together with unlike poles touching. How much force must be applied to sep- arate them? (b) Figure 11.39.

Fig: 1