Question

Using Maxwell’s equations in free space, derive the wave equation for the

y-component of the electric field vector.

Answer 1

Maxwell’s equations in free space are given by

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

The last two equations give

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

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

or

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

Using identity

$$\nabla\times \nabla\times {\bf E}=\nabla(\nabla\cdot {\bf E})-\nabla^2{\bf E}$$

and first Maxwell’s equation, we obtain

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

Finally

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

Also we have

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