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Please helpppp

  1. Nov 25, 2008 #1
    Let respectively b = (b1, b2, b3) and e = (e1, e2, e3) denote the magnetic
    and electric field in some medium. They are governed by Maxwell’s equations which look as follows:
    (0.1) [tex]\partial[/tex]te = curl b
    (0.2) [tex]\partial[/tex]tb = − curl e
    (0.3) div e = 0
    (0.4) div b = 0.
    Show that each bi and each ei satisfies the wave equation
    ([tex]\partial[/tex]t2 - [tex]\Delta[/tex]) [tex]\varphi[/tex]=0
     
  2. jcsd
  3. Nov 25, 2008 #2

    Dick

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    Try it. It's not that hard. It might help to know curl(curl(A))=grad(div(A))-laplacian(A).
     
  4. Nov 26, 2008 #3
    taking curl of both side:
    curl (dt e) = grad (div e) - laplacian (e)
    curl (dt e) = grad (0) - laplacian (e)

    i dont know what i'm doing =(
     
  5. Nov 26, 2008 #4

    Dick

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    dt e=curl b. Taking curl of both sides. curl(dt e)=curl(curl(b))=grad(div B)-laplacian(B)=grad(0)-laplacian(b)=-laplacian(b). Now curl(dt e)=dt(curl e). Continue...
     
  6. Nov 26, 2008 #5
    curl (dt e) = -laplacian (b)
    dt (curl e) = -laplacian (b)
    -dt b = -laplacian (b)
    dt b = laplacian (b)
    0 = laplacian (b)

    and then taking curl of 0.2:
    dtb = -curl e
    curl (dt b) = curl (-curl e)
    dt (curl b) = -curl (curl e)
    dt (e) = -grad( div e) - laplacian (e)
    = -grad (0) - laplacian (e)
    dt (e) = - laplacian (e)
    0 = laplacian e

    how does that satisfy the wave equation? is it because that they both equal to zero?
     
  7. Nov 26, 2008 #6

    Dick

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    You aren't being very careful with signs and you are dropping a dt. dt(dt(e))=[itex]\partial^2_t E[/itex]. Why are you turning dt(e) and dt(b) into zero? You want to get dt(dt(b))=laplacian(b) and the same for e.
     
  8. Nov 26, 2008 #7
    i finally got down to: dt(dt(b)) = laplacian (b) and dt(dt(e))= -laplacian e

    i still dont understand how that applies to the wave equation.
     
  9. Nov 26, 2008 #8

    Dick

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    You've STILL got a sign wrong in the e part. What you've quoted in the problem as the wave equation is what we've been writing as dt(dt(phi))-laplacian(phi)=0. Look up those symbols.
     
  10. Nov 26, 2008 #9
    i found out where i made a mistake with that negative sign. i got it now. thank you so so muchhhhh!!! happy thanksgiving!
     
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