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Finding the curl of B

  1. Apr 24, 2013 #1
    1. The problem statement, all variables and given/known data

    The electric and magnetic fields in a plane wave propagating in free-space in the z-direction can we represented by (in complex-exponential notation)

    E(x, y, z, t) = E_0 e^i(kz−wt+d ) and B(x, y, z, t) = B_0e^i(kz−wt+d )

    Starting with the Ampère-Maxwell law, ∇ x B =μ_0 ε_0 dE/dt , show that E_0 = −c(z×B0).


    2. Relevant equations



    3. The attempt at a solution

    I'm only having problems when I do curl B (differentiating E is no problem). I'm using the matrix method, but by just looking, it seems like it should be zero since B is in the z direction and there's no field in the x or y direction i.e

    x_______y________z

    d/dx___d/dy_____d/dz

    0_______0____B_0e^i(kz−wt+d )

    I should be getting something like ike^(...), but I'm not. Any insight would be helpful

    Thanks
     
    Last edited by a moderator: Apr 24, 2013
  2. jcsd
  3. Apr 24, 2013 #2

    jedishrfu

    Staff: Mentor

    While B is in the z direction only, the curl B will have components in the x and y directions and not the z direction. So you should review the curl definition again and see where you went astray.
     
  4. Apr 24, 2013 #3

    vela

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    The wave is propagating in the z-direction. The magnetic field doesn't point in the z-direction.
     
  5. Apr 24, 2013 #4
    Sorry, what I meant was from doing the calculation, I found the answer to be zero. That's what I found confusing. It shouldn't be zero.

    Am I assuming wrong that d/dx and d/dy of B will come out as 0 since it's differentiating constants as there are no x or y variables in the field?

    Basically, this is what I got from doing curl B:

    x_hat[d/dy e^(...) - d/dz (0)] -y_hat[d/dx e^(...) - d/dz(0)] + z_hat[0-0]
     
    Last edited by a moderator: Apr 24, 2013
  6. Apr 24, 2013 #5
    Ahh, of course, they're supposed to be orthogonal. I was under the impression that the magnetic field was in the z direction.
     
    Last edited by a moderator: Apr 24, 2013
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