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Generic Local Electromagnetic field - MTW Ex 4.1

  1. Sep 28, 2013 #1

    TerryW

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    Can anyone help me with Ex 4.1 in MTW?

    What is a 'generic' field? My expectation is that it would comprise an Electric Field, with arbitrary direction, a Magnetic Field, also with arbitrary direction, and a radiation field (E and B of equal magnitude) , radiating in an arbitrary direction.

    Trying to put this together however to get the Poynting density of energy flow and the density of energy is a bit of a mess however, so I suspect my expectation is wrong.

    Moving on to the next bit of the problem, if you do define the unit vector n and velocity parameter α as shown, then a solution to this is a radiation field with E along the x axis, B along the y axis, producing a Poynting vector along the z axis. If you then translate to the rocket frame, E X B disappears because B becomes zero, not because B becomes parallel to E. I can't see how B ends up parallel to E.

    Can anyone help?


    TerryW
     
  2. jcsd
  3. Sep 28, 2013 #2

    Bill_K

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    No, just the first two - an arbitrary E field and an arbitrary B field. The "radiation field", more commonly called a null field, is just a special case.

    This is one of the three special cases that you were told NOT to consider. Excluding these three cases, you'll get E and B parallel and both nonzero.
     
  4. Sep 28, 2013 #3

    WannabeNewton

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    A generic EM field is simply one which, relative to a given (in this case inertial) frame, splits into an arbitrary E field and an arbitrary B field.

    Why do you think the B field will become zero once we boost to the rocket frame in general i.e. why should the ##\alpha## parameter for the boost velocity necessarily take us to the rest frame of the source of this EM field?
     
  5. Sep 29, 2013 #4

    TerryW

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    Hi Bill_K and WannabeNewton,

    Thanks for clearing up what the generic field is. I've re-read MTW and can see it now but I don't feel it is totally clear (unusually for MTW!)

    As for thinking that the B field will become zero, I noticed that a solution to the equation for the ratio of energy flow to energy density is Bxsinhα and Eycoshα with E = B,
    giving an E X B in the z direction. So when you translate to the rocket frame travelling along the z axis, B becomes zero. At which point I realised I was well off track.

    I've already had a go at resolving the problem based on what you have told me. I haven't cracked it yet and will drop another post if I get terminally stuck.

    I appreciate your help so far and hope you will be on hand if I need further help.


    Regards


    TeryW
     
  6. Oct 9, 2013 #5

    TerryW

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    Still having problems.

    My attempt at proving that the Electric and Magnetic fields become parallel and aligned to the direction of travel of the rocket is attached in a PDF. I've tried to use the invariants E2 - B2 and the equation for tanh2α n to get some more expressions involving the parallel and perpendicular components of E and B which could be used to reduce my expression for EXB to zero but without success.

    On the other hand, considering the expression EXB = (E2 + B2)tanh2α n, which is a generic expression for a generic filed...I could say that if we were in the rocket frame, then we are now moving with the flow of energy, so α = 0, leading to the required result, but maybe that is a bit of a cheat?


    Regards


    TerryW
     

    Attached Files:

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  7. Jan 8, 2014 #6

    TerryW

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    Sorted

    I've revisited the problem and worked out that my expression for EXB does all cancel out. The first two lines γ[…] and γβ[…] disappear because the parallel components are zero then I use the tanh2α formula to show that γ2[…] + γ2β[…..] - γ2β2[….] reduces to zero.
     
  8. Jan 8, 2014 #7

    WannabeNewton

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    That's awesome Terry! I'm glad it worked out.
     
  9. Jan 8, 2014 #8

    TerryW

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    Looking at my original manuscript, if I recognise early on that the parallel components of E and B are zero, I can get to my last expression for ExB more or less straight away!

    Much tidier!

    TerryW
     
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