Question

Using Maxwell’s equations in free space, derive the wave equation for the z-component of
the electric field vector.

Answer 1

The Maxwell's equations in free space

$$\nabla\times {\bf E}=-\frac{1}{c}\frac{\partial {\bf B}}{\partial t}$$ $$\nabla\times {\bf B}=\frac{1}{c}\frac{\partial {\bf E}}{\partial t}$$ $$\rm div{\bf E}=0$$ $$\rm div{\bf B}=0$$

From the second equation we obtain

$$\nabla\times \frac{\partial {\bf B}}{\partial t}=\frac{1}{c}\frac{\partial^2 {\bf E}}{\partial t^2}$$

Plug first equation into last, we get

$$-c\nabla\times (\nabla\times {\bf E})=\frac{1}{c}\frac{\partial^2 {\bf E}}{\partial t^2}$$

Since

$$\nabla\times (\nabla\times {\bf E})={\rm grad(div {\bf E})}-\nabla^2 {\bf E} =-\nabla^2 {\bf E}$$

we finally obtain

$$\nabla^2 {\bf E}-\frac{1}{c^2}\frac{\partial^2 {\bf E}}{\partial t^2}=0$$

For the z-component of the electric field vector the wave equation

$$\nabla^2 {E_z}-\frac{1}{c^2}\frac{\partial^2 {E_z}}{\partial t^2}=0$$