- #1
charlief
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Homework Statement
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##.
Homework Equations
##\nabla \times \mathbf{E} = -\frac{\partial \mathbf{B}}{\partial t}##
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