St. Vincent College
PH 252: Electromagnetic Fields
Final Exam
12/13/2006
1. Our derivation of atomic polarizability assumed a uniform electronic charge density. This time, use
a charge density ρ(r) = kr and show that the polarization depends on E1/2.
a) First, compute the total charge on the electron cloud if it is a sphere with atomic radius a. Also,
solve for kin terms of the total charge qand the atomic radius.
b) Work out the electric field inside the electron cloud a distance dfrom the center of the spherical
electron charge distribution.
c) Use your result from (a) to rewrite the electric field inside the electron cloud in terms of d2and q2.
d) Finally, equate the force on the nucleus from an external electric field to the force on the nucleus,
when it is displaced dfrom the center of the electron cloud, due to the electron cloud and solve for the
polarization.
2. Three long straight wires are arranged at the corners of an equilateral triangle as shown below.
Two have 1 A of current directed out of the page while the third has 2 A of current directed into the
page. Compute the force per unit length on the wire at the apex (top) of the triangle. Also specify the
direction of the force on this wire.
2 m
1 Α
1 Α2 Α
3. Starting with the Law of Biot and Savart, show that the divergence of the magnetic field must be
zero.
4. There is a current flowing through a loop of wire of radius Rsuch that the current d ensity is ~
J=J0ˆ
θ.
a) Compute the magnetic vector potential ~
Aon the axis of the loop at a distance afrom the plane of
the loop. Note that the vector potential will have only one component.
b) Could this result be used to find the magnetic field ~
B? Explain.
a
R
