A non-conducting sphere of radius r = 33.5 cm has a net charge of Q = 5.7 𝜇C uniformly distributed throughout its volume.

Question

A non-conducting sphere of radius r = 33.5 cm has a net charge of Q = 5.7 𝜇C uniformly distributed throughout its volume.

What is the magnitude of the electric field at a distance of one-half its radius from its center?

E =  __ N/C

Answer 1

The charge density is clearly ρ = Q/(4πr ³/3) and by the Gauss Law is proportional to divE. The charge enclosed at a distance d < r is q = (4πd ³/3)ρ. Integrating over the enclosing volume and applying the Gauss Theorem, q is proportional to ∮E•da, where by symmetry E = Eeᵣ and the area element da = daeᵣ are both radially directed, and E can be a function only of d (not direction eᵣ). Therefore ∮E•da = E4πd ².