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Using verify Faraday's law

  1. Dec 30, 2011 #1
    1. The problem statement, all variables and given/known data

    Use the form of the Lorentz transformation matrix R, and the elcectromagnetic tensor F, and using:

    [itex]\frac{\partial F_{\mu\nu}}{\partial x_{\lambda}}[/itex]+[itex]\frac{\partial F_{\nu\lambda}}{\partial x_{\mu}}[/itex]+[itex]\frac{\partial F_{\lambda\mu}}{\partial x_{\nu}}[/itex]=0

    verify Faraday's law

    curl E=-[itex]\frac{\partial\ B}{\partial t}[/itex]

    3. The attempt at a solution

    I tried to find out what the terms in the equation actually are. I found out that [itex]F_{\mu\nu}[/itex] is the Faraday tensor, but what are [itex]F_{\nu\lambda}[/itex], etc? My notes and textbooks really never explains this. I did spend quite a while trying to work out for myself what those terms mean, but I really have no clue.

    Please help.
     
    Last edited: Dec 30, 2011
  2. jcsd
  3. Dec 30, 2011 #2
    Hello,

    As you seem to know, the tensor indicated by [itex]F_{\mu \nu}[/itex] is the electromagnetic field tensor. It contains (as it is usually represented) the three Cartesian components of both the electric and magnetic fields. In classical electrodynamics, it is typically used in an elegant way of representing Maxwell's laws. Those laws can be determined by doing mathematical operations on the field tensor, and that is the goal of this problem: you are supposed to "derive" Faraday's law from the field tensor. At any rate, the idea here is that the subscripts to "F" are the indices for the tensor. But the key idea is that the letters you use to represent those indices are totally arbitrary. [itex]F_{\mu \nu}[/itex], [itex]F_{\nu \lambda}[/itex], and [itex]F_{\lambda \mu}[/itex] ALL represent the same tensor. What is really meant by this equation is that if you choose a particular value for all three indices present, and substitute in the various derivatives and field tensor entries, you will get a true mathematical statement. Try it. Pick, say, [itex]\mu = 1[/itex], [itex]\nu = 2[/itex], and [itex]\lambda = 3[/itex]; insert the values for the electromagnetic field tensor and the appropriate derivatives, and see if you get a true mathematical statement.
     
    Last edited: Dec 31, 2011
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