31.5.5 Connecting E and B

The electric and magnetic fields of an electromagnetic wave both oscillate, but not independently of each other. The two field strengths are related. $E_{y}$ and $B_{z}$ both satisfy the same wave equation, so the traveling waves—just like a wave on a string—are

\begin{array}{l} E_{y} & = & E_{0} sin (k x - ω t) & = & E_{0} sin [2 π (\frac{x}{λ} - f t)] \\ B_{z} & = & B_{0} sin (k x - ω t) & = & B_{0} sin [2 π (\frac{x}{λ} - f t)] \end{array}

(31.37)

where $E_{0}$ and $B_{0}$ are the amplitudes of the electric and magnetic portions of the wave and, as for any sinusoidal wave, $k = 2 π / λ$ , $ω = 2 π f$ , and $λ f = v = c$ . These waves have to satisfy Equation 31.27; thus

\frac{\partial E_{y}}{\partial x} = \frac{2 π E_{0}}{λ} cos [2 π (\frac{x}{λ} - f t)] = - \frac{\partial B_{z}}{\partial t} = 2 π f B_{0} cos [2 π (\frac{x}{λ} - f t)]

(31.38)

Equation 31.38 is true only if $E_{0} = λ f B_{0} = c B_{0}$ . And since the electric and magnetic fields oscillate together, this relationship between their amplitudes has to hold true at any point on the wave. Thus $E = c B$ at any point on the wave.