(1) By considering the flux of particle number across the surface of a cube of side a and letting , derive the conservation law:
(2) The energy- momentum tensor for a perfect fluid is:
in the momentarily comoving reference frame [ MCRF ]. By applying a Lorentz transformation , show that in a general frame it is given by
Prove that the time and space parts of the conservation equations in the MCRF are
and derive the Newtonian limit of these equations. [ This is just filling in the algebra of the derivation in section 4.1 of the lecture Notes.]
(3) Using the definition of the electromagnetic tensor and the current four- vector given in the lectures, show that Maxwell's equations can be written as
Perform a Lorentz transformation on to a frame with velocity v in the x- direction to prove that the part of the electric field perpendicular to changes to , while the part along is unchanged.
(4) The four- force on a particle of charge q and four- velocity is
Express its components in terms of , and the three- velocity . By writing and using Maxwell's equations, show that
is the energy- momentum tensor of the electromagnetic field. Infer that the energy density is
NOTE: This question is worth a bottle of wine!!!
(5) A particle of charge q and rest mass , moves in a circular orbit of radius R in a uniform B field .
(5a) Find B in terms of R, q, and , the angular frequency.
(5b) The speed of the particle is constant since the B field can do no work on the particle. An observer moving at velocity , however, does not see the speed as a constant. What is measured by this observer?
(5c) Calculate and thus . Explain how the energy of the particle can change since the B field does no work on it.
If you have any problems please come and see me or contact me by email.