# Showing the components of a plane EM wave are perpendicular

## Homework Statement

I've derived the EM wave equations from Maxwell's equations, and I now need to show that the E and B components are both perpendicular to each other and to the direction of propagation.

The text book I've been using attempts to show why this is, but it isn't particularly clear and seems to assume that propagation and one of the components are perpendicular.

## Homework Equations

$$\nabla^2E = \mu_0 \epsilon_0 \ddot{E}$$
$$\nabla^2B = \mu_0 \epsilon_0 \ddot{B}$$

Physics Monkey
Homework Helper
You might start by solving the wave equations for E and B i.e. writing down the most general form for E and B consistent with a wave of definite frequency and wavelength. Try plugging those guesses into the original Maxwell equations and see what you find.

That's kind of what I've tried, but I end up assuming that the E component and direction of propagation are perpendicular.

$$\vec{E} = (E_0, 0, 0) \sin(\omega(t + \frac{z}{v}))$$

$$\vec{\nabla} \times \vec{E} = -\dot{\vec{B}} = (0, E_0, 0) \frac{\omega}{v}\cos(\omega(t + \frac{z}{v}))$$

This shows that E and B are perpendicular, but in doing so I've assumed that E and the direction of the propagation are perpendicular.

I thought about using the Poynting vector to show that the direction of propagation is perpendicular to E and B, but I wasn't sure as to whether this proved it or not.

nicksauce
$$\nabla \cdot \vec{E} = 0$$ (in free space w/ no source) to show that the electric field and the wave vector are perpendicular.