Gauss' Law — Question 6

Question 6

Zero divergence away from a point does not exclude a concentrated source

For every $r>0$, a radial electric field is $\mathbf E=C\hat{\mathbf r}/r^2$, with $C>0$. You may use $\boldsymbol\nabla\cdot\mathbf E=r^{-2}d(r^2E_r)/dr$ wherever the field is smooth.

1. Find the ordinary volume charge density for $r>0$ and the charge enclosed by any sphere centered at the origin.

2. Explain how these results coexist, and determine the net charge in any spherical annulus that excludes the origin.

Question 6 - Solution

Zero divergence away from a point does not exclude a concentrated source

(a) Method and calculation

$$r>0:\quad\boldsymbol\nabla\cdot\mathbf E=\frac1{r^2}\frac{dC}{dr}=0\Rightarrow\rho=0.$$$$\Phi(R)=4\pi R^2\frac C{R^2}=4\pi C\Rightarrow Q_{\mathrm{enc}}=4\pi\epsilon_0C.$$

(b) Interpretation and further calculation

$$Q_{a<r<b}=\epsilon_0[\Phi(b)-\Phi(a)]=0.$$$$Q_{r=0}=4\pi\epsilon_0C.$$The source is concentrated at the singular origin. The smooth-field divergence expression cannot be applied through that singular point as if the field were regular.

Physical check

Every centered sphere encloses the same concentrated charge, regardless of its nonzero radius.