An infinitely long, solid non-conducting rod (cylinder) withcircular cross section of radius a has its axis along the z-axis.It has a non-uniform volume charge density given in cylindricalcoordinates by ρ(s) = C (s/a)^2 ,where C is a positive constant. Inaddition, there is a uniform volume charge density −σ on the outercylindrical shell of radius b, where σ is a positive constant.Region 2 is a vacuum.

For parts (a) through (c), use Gauss’ Law and determine theelectric fields (both magnitude and direction of the electricfield) in

(a) Region 1: inside the inner cylinder (s < a)

(b) Region 2: between the inner and outer cylinders (a < s< b)

(c) Region 3: outside the outer cylinder (s > b)

(d) Is the electric field continuous at each surface? (s = a ands = b surfaces)

(e) What is the electric potential difference ∆Vab between thesurface of the inner cylinder (s = a) and the surface of the outercylinder (s = b)? Which surface has a higher potential? 1.

2) We can model the earth as a solid spherical conductor ofradius RE surrounded by a concentric spherical conducting shellwith inner radius Ri , and outer radius Ro, which is theionosphere. The earth has charge +Q, while the ionosphere has zeronet charge.

Write all answers in terms of Q, RE, Ri , Ro, and 0.

For parts (a) through (c), what is the surface charge densityon

(a) the outer surface of the inner sphere of radius RE?

(b) the inner surface on the spherical shell at radius Ri?

(c) the outer surface of the spherical shell at radius Ro?

For parts (d) through (g), determine the electric field E(r)everywhere in space:

(d) r < RE

(e) RE < r < Ri

(f) Ri < r < Ro

(g) r > Ro

(h) Calculate the energy of the system.

(i) Determine the potential at the center given that thepotential is zero at r = ∞.

(j) Find the capacitance of the earth-ionosphere system assumingthat the ionosphere has net charge −Q instead of zero.

3) Two charges are on the z-axis, charge +q at z = +a and −q atz = −a. (Hint: This is NOT a continuous charge distribution but twodiscrete point charges.)

(a) Find the electric potential V (x, y, z) at a field point r =(x, y, z).

(b) Find the potential V (x, y, 0) at a point (x, y, 0) on thexy-plane.

(c) What is the total electrostatic energy of this system?

(d) Using the result of part (c), find out how much work ittakes to move the charges closer so that their separation is arather than 2a.