Problem 47P

(a) Write an expression for the volume charge density ρ(r) of a point charge q at r ′. Make sure that the volume integral of ρ equals q.

(b) What is the volume charge density of an electric dipole, consisting of a point charge −q at the origin and a point charge +q at a?

(c) What is the volume charge density (in spherical coordinates) of a uniform, infinitesimally thin spherical shell of radius R and total charge Q, centered at the origin? [Beware: the integral over all space must equal Q.]

a.)

Step 1 of 3

We have to write an expression for the volume charge density ρ(r) of a point charge q at r ′.

The volume charge density ρ(r) of a point charge q at r ′ is given by the expression,

Making sure that the volume integral of equals .

Using Dirac delta function

If contains

if does not contain

Hence,

Therefore, the volume integral of equals .

b.)