Problem

(a) Using Eq. 3.103, calculate the average electric field of a dipole, over a spherical...

(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