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

`\rho = \rho_0 r`

Step-by-step explanation:

As we know by Gauss law

`\int E. dA = (q)/(\epsilon)`

here we know that

`E = (\rho_0 r^2)/(4\epsilon)`

so here we have

`((\rho_0 r^2)/(4\epsilon))(4\pi r^2) = ((\int\rho dV))/(\epsilon)`

now we have

`(\pi \rho_0 r^4)/(\epsilon) = ((\int\rho dV))/(\epsilon)`

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

now differentiate both sides by volume

`(d(\pi \rho_0 r^4))/(dV) = \rho`

`(\pi \rho_0 4r^3 dr)/(4\pi r^2 dr) = \rho`

`\rho = \rho_0 r`