Physics 415: Electromagnetic Theory I
Prof. S. Teitel email@example.com ----- Fall 2004
Problem Set 2
Due Wednesday, October 6, in lecture
- Problem 1 [10 points]
Consider a plane polarize electromagnetic wave described by the vector and scalar
A(r,t) = Aoei(
k·r - t)
(r,t) = oei(
k·r - t)
where the orientation of the vector Ao is arbitrary.
a) Using Maxwell's equations, find the relationship that must hold between
Ao and o.
b) Using the principal of gauge invariance, show that one can transform to a
new but physically equivalent vector potential which is transversely polarized,
i.e. Ao·k = 0. Explicitly find the
gauge transformation function that does this.
c) What is the scalar potential in the gauge of part (b)?
- Problem 2 [10 points]
Prove Green's reciprocation theorem: If is the potential due to a volume charge density within a volume V and a surface charge density on the conducting surface S bounding the volume V, while ´ is the potential for the same geometry but for a different ´ and ´, then
V d3r ´
+S da ´ =
V d3r ´
+S da ´
- Problem 3 [10 points]
Two infinite grounded parallel conducting planes are separated by a distance d. A point charge q is placed between the planes. Use the reciprocation theorem of Green to prove that the total induced charge on one of the planes is equal to (-q) times the fractional perpendicular distance of the point charge from the other plane. (Hint: As your comparison electrostatic problem with the same surfaces choose one whose charge densities and potential are known and simple.)
- Problem 4 [10 points]
Using the method of images, dicuss the problem of a point charge q inside a hollow, grounded, conducting sphere of inner radius a. Find
a) the potential inside the sphere;
b) the induced surface charge density; what is the total induced charge?
c) the magnitude and direction of the force acting on q. Does q get pushed towards the center, or away from the center?
d) Is there any change in the solution if the sphere is kept at a fixed potential o? If the sphere has a fixed total charge Q?