Using Eq. 5.88, calculate the average magnetic field of a dipole over a sphere of radius R centered at the origin. Do the angular integrals first. Compare your answer with the general theorem in Prob. 5.59. Explain the discrepancy, and indicate how Eq. 5.89 can be corrected to resolve the ambiguity at r = 0. (If you get stuck, refer to Prob. 3.48.)
B _{\text {dip }}( r )=\frac{\mu_{0}}{4 \pi} \frac{1}{r^{3}}[3( m \cdot \hat{ r }) \hat{ r }- m ] (5.89)
Evidently the true field of a magnetic dipole is ^{29}
B _{ dip }( r )=\frac{\mu_{0}}{4 \pi} \frac{1}{r^{3}}[3( m \cdot \hat{ r }) \hat{ r }- m ]+\frac{2 \mu_{0}}{3} m \delta^{3}( r ) (5.94)
Compare the electrostatic analog, Eq. 3.106.
E _{\text {dip }}( r )=\frac{1}{4 \pi \epsilon_{0}} \frac{1}{r^{3}}[3( p \cdot \hat{ r }) \hat{ r }- p ]-\frac{1}{3 \epsilon_{0}} p \delta^{3}( r ) (3.106)
