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Derivation of the wave equation satisfied by E and B

  1. Feb 2, 2015 #1
    1. The problem statement, all variables and given/known data

    given a medium in which p=0, j=0 but where the polarization vector P=P(r,t). Derive the wave equation satisfied by E and B.





    2. Relevant equations
    i started with the 4 basic Maxwells equations
    ∇ · D = ρ (1)
    ∇ · B = 0 (2)
    ∇ × E = −∂B/∂t (3)
    ∇ × H = J + ∂D/∂t (4)

    and with the relation
    D = ɛE + P (5)
    H = 1/µB + M (6)



    3. The attempt at a solution

    i took the curl of both sides of 3 and simplified both sides and got the eqaution
    laplacian of E = µ∂H/∂t (7)
    i assumed that M=0 --> B=µH
    taking the curl of both sides of (7) and using (4)
    curl of (laplacian of E) = (µ∂/∂t)∂(∇ × D)/∂t (8)

    substituting (5) to (8)
    laplacian of E = (µɛ∂/∂t)∂E/∂t + µ∂/∂t)∂P/∂t (9)


    my final answer is for the wave equation of E is (9), i want to know if my answer is correct before trying to solve B


    thank you in advance
     
  2. jcsd
  3. Feb 2, 2015 #2
    That doesn't look right, how did you "simplify" the curls?
     
  4. Feb 3, 2015 #3
    hm.. i might have written the wrong thing in my post but here is my handwritten solution
     

    Attached Files:

  5. Feb 3, 2015 #4
    Yeah there's a problem because the divergence of E is not zero here, only the divergence of D...
     
  6. Feb 3, 2015 #5
    yes thank you

    i reworked that part of the solution

    divergene of E = (1/epsilon) divergence of (D - P)
    = (1/epsilon) {divergence of D - divergence of P}
    = 1/epsilon (p_free - (-p_bound))
    since p_free + p_bound = p , and it is given that p = 0
    thus divergence of E = 0

    is that the only correction?
     
  7. Feb 3, 2015 #6
    Well, your statement says that you are given a medium in which ρ = 0 but in general the charge density you are given is the free one (i.e. ρfree) because this is the one you can control in an experiment.
    This would make ∇⋅E = –∇⋅P/ε = ρbound≠ 0
     
  8. Feb 3, 2015 #7
    oh ok i get it now. ill rework my whole solution
     
  9. Feb 3, 2015 #8

    thank you very much by the way
     
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