Answer to Question #211532 in Electricity and Magnetism for Rocky Valmores

Question

Find the electric field inside a sphere which carries a charge density proportional to the distance from the origin, ρ = kr, for some constant k. (note that this charge density is not uniform, and you must integrate to get the enclosed charge.)

Answer 1

We need to find the electric field inside a sphere which carry charge density of ρ = kr. Consider a spherical Gaussian surface with radius of s and length of l.

Apply Gauss’e law

$\oint E \cdot dA = \frac{Q_{enc}}{e_0}$

the electric field is constant throughout the Gauss’s surface, so we can pull the electric field put of the integral, and the integration over the surface equals the surface area is $4 \pi r^2$

$E \cdot 4 \pi r^2 = \frac{Q_{enc}}{e_0}$

the enclosed charge equals the integration of the density over the volume of the surface

$Q_{enc}= \int ρdτ \\ = \int^{2 \pi}_{0} \int^{\pi}_{0} \int^{r}_{0} (k \bar{r})(\bar{r}^2 sinθd \bar{r}dθd \phi) \\ = 4k \pi \int^{r}_{0} \bar{r}^3 d \bar{r} = \frac{4 \pi k r^4}{4} \\ = \pi k r^4 \\ E \cdot 4 \pi r^2 = \frac{\pi k r^4}{e_0} \\ E = \frac{1}{4 \pi e_0} \pi k r^2 \hat{r}$

where the direction of the field is radially outward from the sphere.