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I'm trying to calculate the Amplitude of electric field in the EM waves using the differential forms from maxwells equations. I've been given frequency ($10^8$ Hz) and displacement current density ($10^{-5} \frac{A}{m^2}$) Useful equations: $c = \lambda f$, $\omega = 2\pi f$, $k = \frac{2\pi}{\lambda}$

and

I've tried: $\nabla \times E = -\frac{\partial{B}}{\partial{t}}$ and $\nabla \times B = \mu_{0} \epsilon_{0} \frac{\partial{E}}{\partial{t}} = \mu_0 10^{-5}$, since $\epsilon_{0} \frac{\partial{E}}{\partial{t}} = 10^{-5}$

I found

$\nabla \times E = \frac{\partial{E}}{\partial{x}}$

which shows

$\frac{\partial{E}}{\partial{x}} = -\frac{\partial{B}}{\partial{t}}$

So, i've tried many different ways to go about this but I get nowhere.

I usually end up with something like:

$E_0 = \frac{B_0}{c}$

But since i don't know what either of the amplitudes are it is useless.

My question is simply how do I go about this?

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There's an easy way and a hard way. I'll start with the easy way. All that you're missing is taking note of what we expect a plane wave to look like: $$\psi(\textbf{r},t)=Ae^{i(\textbf{k}\cdot\textbf{r}-\omega t)}.$$ Applying this to the electric field, we have $$\left|\frac{\partial\textbf{E}}{\partial{t}}\right|=\omega\left|\textbf{E}\right|.$$ Then using your equations, you should be able to get an expression for $|\textbf{E}|$ and $|\textbf{B}|$.

The hard way is to derive why we get to use the equation we start with above. I won't go through everything since you can find this in an introductory text or with google, but the start is taking the curl of one of the curl equations, then substituting in the other. This should eventually lead you to the $\textit{wave equation}$, which is easily solved with separation of variables which in turn lands you at the above equation.

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