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Homework Help: Electric and magnetic field problems (curl/divergence)

  1. Apr 29, 2013 #1
    1. The problem statement, all variables and given/known data

    Consider the electric field E(t,x,y,z) = Acos(ky-wt)k

    1. Find a magnetic field such that [itex]\partial_t[/itex]B + [itex]\nabla[/itex] X E = 0
    2. Show that [itex]\nabla[/itex] . E = 0 and [itex]\nabla[/itex]. B = 0
    3. Find a relationship between k and w that enables these fields to satisfy

    [itex]\nabla[/itex] X B = [itex]\mu_{0}[/itex][itex]\epsilon_{0}[/itex][itex]\frac{\partial E}{\partial t}[/itex]

    3. The attempt at a solution

    Really the problem here is the first one. I understand (sort of) the curl operator, but how do you find [itex]\nabla[/itex] X E? Would you start with a matrix of

    [i j k
    0 kAcos(ky-wt) 0
    0 0 Acos(ky-wt)]

    Then find the determinant, which is Acos(ky-wt)(1+k)i - 0j + 0k?
    Last edited: Apr 29, 2013
  2. jcsd
  3. Apr 29, 2013 #2
    No, that matrix is not correct. The cross product of the del operator [itex] \nabla [/itex] and a vector function is just an alternate convention for denoting curl. You can why here: http://en.wikipedia.org/wiki/Del#Curl. This should also explain why divergence can be denoted: [itex]\nabla \cdot [/itex]

    So your matrix should be: [itex] \left[ \begin{array}{ccc}
    \hat{i} & \hat{j} & \hat{k} \\
    \frac{\partial}{\partial x} & \frac{\partial}{\partial y} & \frac{\partial}{\partial z}\\
    0 & 0 & {\scriptsize A\cos(ky-wt)} \end{array} \right][/itex]

    The rest of the problem should be relatively trivial once you know what the operators do.
    Last edited: Apr 29, 2013
  4. Apr 29, 2013 #3
    Ah I see. Thanks for your help.
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