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Derive the wave equation for fields E, B from the potentials

  1. Feb 14, 2017 #1
    • Thread moved from the technical forums, so no Homework Template is shown
    I'm studying for my electrodynamics exam and one of the past exam questions is:

    From the scalar and vector potentials, derive the homogenous wave equations for E and B fields in vacuum.

    I did derive the wave equation for the B field by simply taking the curl of the homogenous wave equation for the vector potential A. But I got stuck deriving the wave equation for the E field. I think it has to come from a tricky combination of the following four equations:
    1) Definition of the E field E = -Φ - ∂A/dt
    2&3) Wave equations for Φ and A
    4) Lorenz gauge condition: ∇.A - (1/c^2)(∂^2Φ/∂t^2) = 0

    Any help is appreciated!
  2. jcsd
  3. Feb 14, 2017 #2


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    If Φ satisfies the wave equation, can you show that -Φ also satisfies the wave equation?
  4. Feb 14, 2017 #3
    Hmm, so I write -(ΔΦ-(1/c^2)(∂^2Φ/∂t^2))=Δ(-Φ)-(1/c^2)(∂^2(-Φ)/∂t^2)=0. And this operation (i.e, taking nabla 'inside' the Laplacian and the partial time derivative) is allowed, right? There are no restrictions on it as far as I know but I'm not 100% sure.

    Then I perform the same trick for -∂A/∂t. Similarly, I take the partial derivative inside (again, assuming it's allowed), showing the equation is satisfied for -∂A/∂t. Then add up the new equations, use formula 1) above and done?
  5. Feb 14, 2017 #4


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  6. Feb 14, 2017 #5
    That was what I wasn't 100% sure. Thanks!
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