(a) Prove that the average magnetic field, over a sphere of radius R, due to steady currents inside the sphere, is
B _{ ave }=\frac{\mu_{0}}{4 \pi} \frac{2 m }{R^{3}} , (5.93)
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:
B _{ ave }=\frac{1}{\frac{4}{3} \pi R^{3}} \int B d \tau .
E _{ ave }=-\frac{1}{4 \pi \epsilon_{0}} \frac{ p }{R^{3}} (3.105)
Write B as (∇ × A), and apply Prob. 1.61(b). Now put in Eq. 5.65, and do the surface integral first, showing that
\int \frac{1}{ᴫ} d a =\frac{4}{3} \pi r ^{\prime} .
A ( r )=\frac{\mu_{0}}{4 \pi} \int \frac{ J \left( r ^{\prime}\right)}{ᴫ} d \tau^{\prime} (5.65)
(see Fig. 5.65). Use Eq. 5.90, if you like.]
m =\frac{1}{2} \int( r \times J ) d \tau (5.90)
(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.
