Name___________________
(1) [18 points] Two long cylindrical conducting shells with radius a and b are arranged
concentrically, with a < b, and are charged so that their potentials are Va and Vb, respectively.
(a) Write down the general form of the electric field between the shells.
(b) Find the potential at points between the shells.
(c) Now supposing that Vb = 0, but that Va remains finite, find the energy stored in a length L of
this system, in terms of Va and the system dimensions L, a, and b.
2
Name___________________
! !
!
(2) [10 points] (a) Find the electric field corresponding to the electric potential, V = k ! r , with k
a fixed vector, and give a physical example of a situation that could produce such a field.
(b) For each of the following two electric fields, determine whether it can exist in electrostatics,
and if it can exist, determine a charge density that could produce it. ! and ! are constants.
!
!
(i) E = ! (xŷ " yx̂) ; (ii) E = ! (xx̂ + yŷ)
!
(3) [10 points] (a) Find the magnetic field corresponding to the vector potential A = !b ln(s)ẑ
(in cylindrical coordinates; b is constant) and give an example of a physical situation that would
produce this field.
(b) For each of the following magnetic fields, determine whether it can exist in magnetostatics,
and if it can exist, determine a current density that could produce it. ! and ! are constants.
!
!
(i) B = ! (xŷ " yx̂) ; (ii) B = !(xx̂ + yŷ)
3
Name___________________
(4) [17 points] A long grounded conducting cylindrical rod with radius a is placed in a uniform
electric field, of magnitude Eo, directed in the x direction, perpendicular to the axis of the
cylinder.
(a) Find the electric field at points outside the cylinder.
(b) Find the charge density vs. angle on the surface of the rod.
4
Name___________________
(5) [18 points] A sphere of ferromagnetic material, centered about the origin, has a uniform
magnetization, M o ẑ , with M o a constant. The sphere has radius R.
(a) Find the bound currents.
(b) Find the total dipole moment of this sphere.
! µ M R3
2 cos! r̂ + sin !!ˆ , find the
(c) Given that the magnetic field outside the sphere is B = o o3
3r
magnetic field at points just inside the sphere, showing how this field is derived by using the
appropriate boundary conditions. Give the final solution in terms of x̂ , ŷ , and ẑ (and note that
this should simplify the answer).
(
5
)
Name___________________
(6) [17 points] A parallel-plate capacitor has area A and plate separation d. The lower plate
carries free charge +Q, and sits on the x-y plane at z = 0. The upper plate sits at z = d and has free
charge –Q. The interior of the capacitor is filled with a dielectric whose relative permittivity
changes with z according to, ! r = ez /a , with a being a constant.
!
(a) Find the displacement field, D , inside the capacitor.
(b) Find the electric field in the capacitor, and the potential difference between the plates.
(c) Find the surface and volume bound charge densities in the dielectric.
6
Name___________________
(7)
! [102points] A long wire, radius R, carries a current along its length with current density,
J = as ẑ .
(a) Find the total current carried by the wire.
(b) Using Ampere’s law find the magnetic field, with direction, both for points s < R and
s > R.
7