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Homework Help: Find Magnetic Field from Electric Field Using Maxwell's Equations

  1. Jul 7, 2014 #1
    1. The problem statement, all variables and given/known data
    An electromagnetic wave has an electric field [tex]\mathbf{E} = E_0 \cos(kz-ωt) \hat{x}[/tex]. Using Maxwell's equations, find the magnetic field.

    2. Relevant equations
    [tex]\mathbf{∇\times E} = \mathbf{\dot{B}}[/tex]

    3. The attempt at a solution

    So this problem appears extremely simple, but other students have told me my answer is incorrect, and I can't figure out what is wrong with my math. I find the cross product, which results in the following equation for the time derivative of the magnetic field:

    [tex]\mathbf{\dot{B}} = \hat{y}kE_0\sin(kz-ωt)[/tex]

    I now integrate both sides with respect to time. This is where my answer diverges from others, so I'll fully write out my steps:

    [tex]\mathbf{B} = kE_0∫_0^t \sin(kz-ωt') dt'\hat{y} [/tex]

    I set [tex]u = kz-ωt'[/tex], which means [tex]du = -ωdt'[/tex]

    Plugging this in, the new integral is:

    [tex]\mathbf{B} = -\frac{kE_0}{ω}∫_{kz}^{kz-ωt}\sin(u)du[/tex]

    The result is then

    [tex]\mathbf{B} = \frac{kE_0}{ω}[\cos(kz-ωt)-\cos(kz)]\hat{y}[/tex]

    However, every student I've talked to has told me that the correct answer should be

    [tex] \mathbf{B} = \frac{kE_0}{ω} \cos(kz-ωt)\hat{y} [/tex]

    Is there something simple I'm missing? There's nothing else in the problem description I didn't write. The other answer looks more correct but I can't find any reason that mine is incorrect.
    Last edited: Jul 7, 2014
  2. jcsd
  3. Jul 7, 2014 #2


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    The left side of the second equation above is incorrect. Think again about what you get when you integrate the left side of the first equation from t' = 0 to t' = t.
    Last edited: Jul 7, 2014
  4. Jul 8, 2014 #3
    Woops thanks for the help, figured it out!
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