Tensors and Relativity: Assignment 4

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 tex2html_wrap_inline435 , derive the conservation law:

    eqnarray379

    Hint!

     

  • (2) The energy- momentum tensor for a perfect fluid is:

    eqnarray381

    in the momentarily comoving reference frame [ MCRF ]. By applying a Lorentz transformation , show that in a general frame it is given by

    eqnarray383

     

    Prove that the time and space parts of the conservation equations tex2html_wrap_inline437 in the MCRF are

    eqnarray385

    and

    eqnarray387

    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.]

    Hint!

     

  • (3) Using the definition of the electromagnetic tensor tex2html_wrap_inline439 and the current four- vector tex2html_wrap_inline441 given in the lectures, show that Maxwell's equations can be written as

    eqnarray391

    and

    eqnarray393

     

    Perform a Lorentz transformation on tex2html_wrap_inline443 to a frame tex2html_wrap_inline445 with velocity v in the x- direction to prove that the part of the electric field perpendicular to tex2html_wrap_inline449 changes to tex2html_wrap_inline451 , while the part along tex2html_wrap_inline449 is unchanged.

    Hint!

     

  • (4) The four- force on a particle of charge q and four- velocity tex2html_wrap_inline457 is

    eqnarray406

    Express its components in terms of tex2html_wrap_inline459 , tex2html_wrap_inline461 and the three- velocity tex2html_wrap_inline463 . By writing tex2html_wrap_inline465 and using Maxwell's equations, show that

    eqnarray418

    where

    eqnarray420

    tex2html_wrap_inline467 is the energy- momentum tensor of the electromagnetic field. Infer that the energy density is

    eqnarray422

    Hint!

    NOTE: This question is worth a bottle of wine!!!

     

  • (5) A particle of charge q and rest mass tex2html_wrap_inline471 , moves in a circular orbit of radius R in a uniform B field tex2html_wrap_inline477 .

    (5a) Find B in terms of R, q, tex2html_wrap_inline471 and tex2html_wrap_inline487 , 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 tex2html_wrap_inline491 , however, does not see the speed as a constant. What is tex2html_wrap_inline493 measured by this observer?

    (5c) Calculate tex2html_wrap_inline495 and thus tex2html_wrap_inline497 . Explain how the energy of the particle can change since the B field does no work on it.

    Hint!

     

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