Question

Consider a sphere of radius R with charge density distributed as

ρ (r ) = kr for r ≤ R

= 0 for r > R .

(a) Find the electric field at all points r. 

(b) Suppose the total charge on the sphere is 2e where e is the electron charge. Where can two protons be embedded such that the force on each of them is zero. Assume that the introduction of the proton does not alter the negative charge distribution.

Answer 1

The symmetry of the problem suggests that the electric field is radial. For points r < R, consider spherical Gaussian surfaces. Then on the surface

For points, r > R, consider a spherical Gaussian surfaces’ of radius r,

The two protons must be on the opposite sides of the centre along a diameter. Suppose the protons are at a distance r from the centre.

Consider the forces on proton 1. The attractive force due to the charge distribution is

Thus, the protons must be at a distance r =R/4√8 from the centre.