Finding the Frequency Domain and Time Domain magnetic field

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To find the frequency and time domain magnetic fields, apply Faraday's law in differential form, which relates the curl of the electric field to the time derivative of the magnetic field. By calculating the curl of the known electric field and integrating with respect to time, the magnetic field can be derived. It's noted that the expression for the magnetic field may require a factor of 1/c, indicating a relationship between the electric and magnetic fields in a plane wave. Understanding the relationship between the electric field, magnetic field, and wave propagation direction is essential for solving the problem. A clearer explanation of these concepts can significantly aid in grasping the solutions.
EmmanKR
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I was wondering if anyone could walk me though a better explanation on how to get the given results for these two questions. The solutions posted by my professor aren't really clear to me so if anyone is able to better explain how to get the solution it would be much appreciated!
 

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For c) You should use Faraday's law in differential form $$\nabla\times\mathbf{E}=-\frac{d\mathbf{B}}{dt}\Rightarrow \mathbf{B}=-\int \nabla\times\mathbf{E} dt$$
You know E in explicit form so you can calculate its curl and then calculate the time integral to get B.
Your handwriting is pretty bad but I think your expression for H is missing a ##\frac{1}{c}## factor.
 
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You should be able to answer part (c) almost by inspection. You have a plane wave in free space. How are the magnetic field, the electric field, and the direction the wave propagates related?
 
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