Problem

Using Eq. 5.88, calculate the average magnetic field of a dipole over a sphere of radius...

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.) Evidently the true field of a magnetic dipole is²⁹

Compare the electrostatic analog, Eq. 3.106.

Reference equation 5.88

Reference equation 5.89

Prob. 5.59

(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:

Reference prob 3.48

(a) Using Eq. 3.103, calculate the average electric field of a dipole, over a spherical volume of radius R, centered at the origin. Do the angular integrals first. [Note: You must express in terms of (see back cover) before integrating. If you don’t understand why, reread the discussion in Sect. 1.4.1.] Compare your answer with the general theorem (Eq. 3.105). The discrepancy here is related to the fact that the field of a dipole blows up at r = 0. The angular integral is zero, but the radial integral is infinite, so we really don’t know what to make of the answer. To resolve this dilemma, let’s say that Eq. 3.103 applies outside a tiny sphere of radius ?—its contribution to E_ave is then unambiguously zero, and the whole answer has to come from the field inside the -sphere.

(b) What must the field inside the ? - sphere be, in order for the general theorem (Eq. 3.105) to hold? [Hint: since ? is arbitrarily small, we’re talking about something that is infinite at r = 0 and whose integral over an infinitesimal volume is finite.]

Evidently, the true field of a dipole is

You may wonder how we missed the delta-function term23 when we calculated the field back in Sect. 3.4.4. The answer is that the differentiation leading to Eq. 3.103 is valid except at r = 0, but we should have known (from our experience in Sect. 1.5.1) that the point r = 0 would be problematic.²⁴

Reference equation 3.105