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I'm trying to do a question which reads:

Perfectly conducting planes are positioned at $y=0$ and $y=a$. Show that a monochromatic wave may propagate between the plates in the direction $z$ if the field components are:

$$E_x = \omega A \sin(\frac{n \pi y}{a})\sin(kz -\omega t)$$ $$B_y=kA\sin(\frac{n\pi y}{a})\sin(kz-\omega t)$$ $$B_z=\frac{n\pi A}{a}\cos(\frac{n\pi y}{a})\cos(kz-\omega t)$$ with $A$ a constant and $n\in \mathbb{Z}$.

It seems to me that, unless I am mistaken, the fields defined above do not satisfy $$\nabla \times \textbf{B} = \epsilon_0 \mu_0 \frac{\partial \textbf{E}}{\partial t}.$$

This can be recitified by having $\textbf{J} \neq \textbf{0}$, is this possible? I can't see how it can be in free space. I'm not entirely sure what to do now.

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It seems to me that, unless I am mistaken, the fields defined above do not satisfy $$\nabla \times \textbf{B} = \epsilon_0 \mu_0\frac{\partial \textbf{E}}{\partial t}.$$

As long as $$ \omega^2\epsilon_0\mu_0=k^2+\frac{n^2\pi^2}{a^2} $$ then they do satisfy the above equation.

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