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From wave equation to maxwell equation

  1. Jul 17, 2011 #1
    in electromagnetic books, we see by the aid of vector calculus, we can reach to wave equation from Maxwell 's equations.

    is it possible to reach to Maxwell 's equations from wave equations?

    in the other word, in electromagnetic books we get Maxwell 's equations as phenomenological principles and drive wave equations from them.
    is it possible to get wave equations as phenomenological principles and drive Maxwell 's equations from them?

    do you try to calculate it?
     
  2. jcsd
  3. Jul 17, 2011 #2

    Bill_K

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    sadegh4137, Yes, this is possible. Start with the two wave equations ∇2A - (1/c2)∂2A/∂t2 = 4πJ and ∇2φ - (1/c2)∂2φ/∂t2 = 4πρ, and the Lorenz gauge condition ∇·A + (1/c)∂φ/dt = 0. Define B ≡ ∇ x A and E ≡ - ∇φ - (1/c)∂A/dt, and you can easily show that E and B satisfy Maxwell's Equations.
     
  4. Jul 17, 2011 #3

    Meir Achuz

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    It just goes through the steps of the derivation of the wave equations from Maxwell's equations backwards. This can be done for most derivations.
     
  5. Jul 17, 2011 #4
    thanks
    why we can define B ≡ ∇ x A and E ≡ - ∇φ - (1/c)∂A/dt ?

    you consider that B and E define by above two equations.
    and these satisfy ME easily, yes you are right.

    but if you consider B and E define by other equations,
    like E ≡ ∇ x A & B ≡ - ∇φ - (1/c)∂A/dt
    these can't satisfy ME!

    it seems you know ME before this and define E & B like this.
    you should assume we have only wave equation and now we want to derive field equation from them.
    like ME, we don't know wave equation.
    by some calculation from ME derive them.
    and now, we have WE, not before this.
     
  6. Jul 17, 2011 #5

    Bill_K

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    Of course if you define E and B some nonstandard way, they won't satisfy Maxwell's Equations. What's your point.
     
  7. Jul 17, 2011 #6
    I don't think you can obtain Maxwell's equations from the wave solutions, because they are just a special case of the Maxwell equations (no charges and currents). So, the Maxwell equations contain more information than the wave equation, that's why you can only go one way in the derivation.
     
  8. Jul 17, 2011 #7
    yes you are right, Sybren
    but if we want to reach to ME in vacuum, I think that we haven't lose any information. ( no charge and current )
    or we want to consider WE in general case with charge and current.



    Bill_K, why do you think that other definitions aren't standard?
    we have WE & we want to field equation.
    we don't know those and want to calculate it.
    you define B & E like this and another people define in other way
    with your definition, you drive some field equation and another person drive another
    now, which one is correct?
     
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