2.14

2.14 Find the electric field inside a sphere which carries a charge density proportional to the distance from the origin, $$\mu = k \rho$$, for some constant $$\rho$$.

This is efficiently solved with Gauss’ law.

$$\oint_S \vec{E} \cdot d\vec{A} = \frac{Q_{enc}}{\epsilon_0}$$

Since the charge density is not uniform integrate to find $$Q_{enc}$$.

$$Q_{enc} = \int_0^r \int_0^{2 \pi} \int_0^{\pi} \mu \rho^2 \sin \phi \, d\theta \, d\phi \, d\rho = 4 \pi \int_0^R k \rho^3 \, d\rho$$

$$Q_{enc} = k \pi r^4$$

$$\left| \left| \vec{E} \right| \right| (4 \pi r^2) = \frac{ k \pi r^4 }{\epsilon_0}$$

$$\left| \left| \vec{E} \right| \right| = \frac{ k r^2}{ 4 \epsilon_0 }$$