Use the charge distribution and electric field calculated (a) Show that for r ≥ R the pote...

Use the charge distribution and electric field calculated (a) Show that for r ≥ R the potential is identical to that produced by a point charge Q. (Take the potential to be zero at infinity.) (b) Obtain an expression for the electric potential valid in the region r ≤ R.

Problem:

A nonuniform, but spherically symmetric, distribution of charge has a charge density  ρ(r) given as follows:

Where  ρ0 = 3Q/πR3 is a positive constant. (a) Show that the total charge contained in the charge distribution is Q. (b) Show that the electric field in the region r ≥ R is identical to that produced by a point charge Q at r = 0. (c) Obtain an expression for the electric field in the region r ≤ R. (d) Graph the electric-field magnitude E as a function of r. (e) Find the value of r at which the electric field is maximum, and find the value of that maximum field.