Question

The electic field inside a spherical volume of radius a is given by: vector E = p_0 r^2 / 4 epsilon r cap Find an expression for the charge density inside the spherical volume that gives rise to this electric field.

Answer 1

Answer:

$\rho = \rho_0 r$

Explanation:

As we know by Gauss law

$\int E. dA = \frac{q}{\epsilon}$

here we know that

$E = \frac{\rho_0 r^2}{4\epsilon}$

so here we have

$(\frac{\rho_0 r^2}{4\epsilon})(4\pi r^2) = \frac{(\int\rho dV)}{\epsilon}$

now we have

$\frac{\pi \rho_0 r^4}{\epsilon} = \frac{(\int\rho dV)}{\epsilon}$

$\pi \rho_0 r^4 = (\int\rho dV)$

now differentiate both sides by volume

$\frac{d(\pi \rho_0 r^4)}{dV} = \rho$

$\frac{\pi \rho_0 4r^3 dr}{4\pi r^2 dr} = \rho$

$\rho = \rho_0 r$