1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

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

    TSny

    User Avatar
    Homework Helper
    Gold Member

    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

    TSny

    User Avatar
    Homework Helper
    Gold Member

  6. Feb 14, 2017 #5
    That was what I wasn't 100% sure. Thanks!
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Have something to add?
Draft saved Draft deleted