Applying Gauss' Law: Spherical Symmetry — Question 9

Question 9

Designing a constant radial field requires density inversely proportional to radius

In a spherically symmetric device, require $E_r=E_0>0$ throughout an insulating layer $a<r<b$. The central region $r<a$ may contain a charge distribution to be chosen. There are no surface sheets at a or b, and no volume charge beyond b.

1. Find the required volume density in the layer and the required total core charge.

2. Find the total charge of the device and its exterior field. Explain why constant radial field does not mean constant Cartesian vector field.

Question 9 - Solution

Designing a constant radial field requires density inversely proportional to radius

(a) Method and calculation

$$Q_{\mathrm{enc}}(r)=4\pi\epsilon_0E_0r^2\Rightarrow4\pi r^2\rho(r)=8\pi\epsilon_0E_0r.$$$$\rho(r)=\frac{2\epsilon_0E_0}{r},\quad Q_{\mathrm{core}}=4\pi\epsilon_0E_0a^2.$$

(b) Interpretation and further calculation

$$Q_{\mathrm{total}}=4\pi\epsilon_0E_0b^2,\quad E_r(r>b)=E_0\frac{b^2}{r^2}.$$$$\mathbf E=E_0\hat{\mathbf r},\quad\hat{\mathbf r}\text{ changes with position}.$$The magnitude is constant in the layer, while the diverging directions still require positive volume charge.

Physical check

The exterior field equals $E_0$ at b, so no normal-field jump or surface sheet is needed there.