Applying Gauss' Law: Spherical Symmetry — Question 7

Question 7

A smoothed Coulomb field reveals an exponential radial charge distribution

A spherically symmetric field is measured as $E_r(r)=Q[1-e^{-(r/a)^3}]/(4\pi\epsilon_0r^2)$ for $r>0$, with $Q,a>0$. Assume no source singularity is hidden at the origin.

1. Infer $Q_{\mathrm{enc}}(r)$ and the volume density using $dQ_{\mathrm{enc}}/dr=4\pi r^2\rho(r)$.

2. Find the radius containing half the total charge and the leading field near the origin. Check regularity there.

Question 7 - Solution

A smoothed Coulomb field reveals an exponential radial charge distribution

(a) Method and calculation

$$Q_{\mathrm{enc}}(r)=4\pi\epsilon_0r^2E_r=Q[1-e^{-(r/a)^3}].$$$$\rho(r)=\frac1{4\pi r^2}\frac{dQ_{\mathrm{enc}}}{dr}=\frac{3Q}{4\pi a^3}e^{-(r/a)^3}.$$

(b) Interpretation and further calculation

$$Q_{\mathrm{enc}}(r_{1/2})=Q/2\Rightarrow r_{1/2}=a(\ln2)^{1/3}\approx.884997a.$$$$r\ll a:\quad1-e^{-(r/a)^3}\simeq(r/a)^3\Rightarrow E_r\simeq\frac{Qr}{4\pi\epsilon_0a^3}.$$The density is finite and the vector field tends continuously to zero at the center.

Physical check

The enclosed charge approaches Q at infinity and approaches zero as $r^3$ at the origin, leaving no point-charge term.