Question

starting from maxwell's equations derive wave equation for a uniform plane wave travelling in free space.

Answer 1

$\nabla E=-\frac{\rho}{\epsilon_0}$

$E=E_0e^{kx-wt}$

$$=\nabla E=\frac{d(E_0e^{kx-wt})}{dx}=kE_0e^{kx-wt}=kE$$

$kE=-\frac{\rho}{\epsilon_0}\\E=-\frac{\rho}{k\epsilon_0}$

Wave equation