Introduction to Electrodynamics · Problem 3.47

Potentials — David J. Griffiths

Physics · average-field-dipole

Problem

Show that the average field inside a sphere of radius RR, due to all the charge within the sphere, is Eave=−14πϵ0pR3,\mathbf{E}_{\text{ave}} = -\frac{1}{4\pi\epsilon_0}\frac{\mathbf{p}}{R^3}, where p\mathbf{p} is the total dipole moment. There are several ways to prove this delightfully simple result. Here's one method: (a) Show that the average field due to a single charge qq at point r′\mathbf{r}' inside the sphere is the same as the field at r′\mathbf{r}' due to a uniformly charged sphere with ρ=−q/(43πR3)\rho = -q/(\frac{4}{3}\pi R^3), namely 14πϵ01(43πR3)∫qr2r^ dτ′,\frac{1}{4\pi\epsilon_0}\frac{1}{(\frac{4}{3}\pi R^3)}\int \frac{q}{\mathscr{r}^2}\hat{\boldsymbol{\mathscr{r}}}\, d\tau', where r\boldsymbol{\mathscr{r}} is the vector from r′\mathbf{r}' to dτ′d\tau'. (b) The latter can be found from Gauss's law (see Prob. 2.12). Express the answer in terms of the dipole moment of qq. (c) Use the superposition principle to generalize to an arbitrary charge distribution. (d) While you're at it, show that the average field over the volume of a sphere, due to all the charges outside, is the same as the field they produce at the center.

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