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Find expression for electric field from magnetic field

  1. Nov 30, 2015 #1
    1. The problem statement, all variables and given/known data
    In a region of space, the magnetic field depends on the co-ordinate ##z## and is given by $$\mathbf{B} = \hat{\jmath} B_0 \cos \left(kz - \omega t \right)$$ where ##k## is the wave number, ##\omega## is the angular frequency, and ##B_0## is a constant.
    The Electric Field in Cartesian coordinates is ##\mathbf{E} = \hat{\imath} E_x + \hat{\jmath} E_y + \hat{k} E_z##. Given that ##E_y = E_z = 0## and ##E_z = \omega B_0/k## at ##z = t = 0##, determine an expression for ##E_x##.

    2. Relevant equations

    ##\nabla \times \mathbf{E} = -\frac{\partial \mathbf{B}}{\partial t}##

    3. The attempt at a solution
    Using the cross product rule, I changed ##\nabla \times \mathbf{E} = -\frac{\partial \mathbf{B}}{\partial t}## to ##-\frac{\partial \mathbf{B}}{\partial t} \times \nabla = \mathbf{E}##. Calculated ## -\frac{\partial \mathbf{B}}{\partial t} = -\hat{\jmath} B_0\omega\sin(kz - \omega t)##. Then working out the cross product I got ##\mathbf{E} = \hat{\imath} \frac{\partial}{\partial z}[-B_0\omega\sin(kz - \omega t)] = -\hat{\imath} B_0 k \omega \cos(k z - \omega t)##. So inputting ##z = t = 0## clearly gives ##B_0 \omega k## instead of ## \omega B_0 / k##.

    I cannot see where in my method I have gone wrong and I am not sure this method is correct?
    Thank you so much
     
  2. jcsd
  3. Nov 30, 2015 #2
    Why do you say that [tex]E_{z}=0[/tex] and then [tex]E_{z}=(\omega B_{0})/k[/tex]. You can't say that the z component of electric field is zero and non zero at the same time... clear that confusion then I'll solve your problem.
     
  4. Nov 30, 2015 #3
    Apologies it was a typo, I meant "##E_y = E_z = 0## and ##E_x = (\omega B_0)/k##"
     
  5. Nov 30, 2015 #4
    Oh.. then you just have to use this;

    [tex]\vec{\nabla}\times \vec{B}=\mu_{0}\vec{J}+\mu_{0}\epsilon_{0}\frac{\partial \vec{E}}{\partial t}[/tex]

    Since you dont have any currents, so first term on the right hand side is zero, so you are left with;

    [tex]\vec{\nabla}\times \vec{B}=\mu_{0}\epsilon_{0}\frac{\partial \vec{E}}{\partial t}[/tex]

    which is easy to solve.
     
  6. Nov 30, 2015 #5
    Thank you very much!!
     
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