(a) Prove that the average magnetic field, over a sphere of radius R, due to steady curr...

(a) Prove that the average magnetic field, over a sphere of radius R, due to steady currents inside the sphere, is

where m is the total dipole moment of the sphere. Contrast the electrostatic result, Eq. 3.105. [This is tough, so I’ll give you a start:

Write B as (∇ × A), and apply Prob. 1.61(b). Now put in Eq. 5.65, and do the surface integral first, showing that

(see Fig. 5.65). Use Eq. 5.90, if you like.]

(b) Show that the average magnetic field due to steady currents outside the sphere is the same as the field they produce at the center.

Equation 5.90

Fig. 5.65.

Reference Prob. 1.61(b).

Although the gradient, divergence, and curl theorems are the fundamental integral theorems of vector calculus, it is possible to derive a number of corollaries from them. Show that: