# Problem

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### Peter Dunsby Room 310a Applied Mathematics

- (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

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
and

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 isExpress its components in terms of , and the three- velocity . By writing and using Maxwell's equations, show that

where

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.**

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