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Physics 415: Electromagnetic Theory I
Prof. S. Teitel stte@pas.rochester.edu ----- 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 potentials,

  A(r,t) = A_oe^{i( k·r - t)}     (r,t) = _oe^{i( k·r - t)}

  where the orientation of the vector A_o is arbitrary.

  a) Using Maxwell's equations, find the relationship that must hold between A_o 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. A_o·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?