Phys 200C – HW5

Due Friday 27-Jan-06 5PM to Duk Shin

5-1. If n is a unit three vector in the direction of propagation of an electro-magnetic wave, then according to classical electromagnetism, . Show that this statement is also invariant under Lorentz transformation by showing its equivalence to the manifestly Lorentz invariant statement , where is the four-vector oriented in the direction of propagation of the wave and is the field strength tensor.

5-2. (Problem 11.14 in Jackson.)

a) Express the Lorentz scalars F^mnF_mn, F^mnF_mn, and F^mnF_mn in terms of E and B. Are there other invariants quadratic in the field strengths E and B?

b) Is it possible to have an electromagnetic field that appears as a purely electric field in one inertial frame and as a purely magnetic field in some other inertial frame? What are the criteria imposed on E and B such that there is an inertial frame in which there is no electric field?

c) For macroscopic media, E, B for the tensor field F^mn and D, H the tensor G^mn. What further invariants can be formed? What are their explicit expressions in terms of the 3-vector fields? (Tensor G^mn is formed from F^mn by letting E -> D and B -> H).