A) Derive a relation between the wavevector (K) and angular frequency (omega) for the electric field to be a solution to the wave-equation:

Here's the solution to question 4:

a) Derive a relation between the wavevector ($\vec{K}$) and angular frequency ($\omega$) for the electric field to be a solution to the wave-equation:

The electric field is given by $\vec{E}(z, t) = E_0 e^{i(kz-\omega t)} \hat{x}$.
The wave equation (as given in the problem for derivation purposes) is $\nabla^2 \vec{E} - \frac{1}{\epsilon_0 \mu_0} \frac{\partial^2 \vec{E}}{\partial t^2} = 0$.

Step 1: Calculate the Laplacian of $\vec{E}$.
Since $\vec{E}$ only has an $