# 16.2 Plane electromagnetic waves  (Page 3/5)

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${E}_{y}\left(x,t\right)=f\left(\xi \right)\phantom{\rule{1.5em}{0ex}}\text{where}\phantom{\rule{0.2em}{0ex}}\xi =x-ct.$

It is left as a mathematical exercise to show, using the chain rule for differentiation, that [link] and [link] imply

$1={\epsilon }_{0}{\mu }_{0}{c}^{2}.$

The speed of the electromagnetic wave in free space is therefore given in terms of the permeability and the permittivity of free space by

$c=\frac{1}{\sqrt{{\epsilon }_{0}{\mu }_{0}}}.$

We could just as easily have assumed an electromagnetic wave with field components ${E}_{z}\left(x,t\right)$ and ${B}_{y}\left(x,t\right)$ . The same type of analysis with [link] and [link] would also show that the speed of an electromagnetic wave is $c=1\text{/}\sqrt{{\epsilon }_{0}{\mu }_{0}}$ .

The physics of traveling electromagnetic fields was worked out by Maxwell in 1873. He showed in a more general way than our derivation that electromagnetic waves always travel in free space with a speed given by [link] . If we evaluate the speed $c=\frac{1}{\sqrt{{\epsilon }_{0}{\mu }_{0}}},$ we find that

$c=\frac{1}{\sqrt{\left(8.85\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}{10}^{-12}\frac{{\text{C}}^{2}}{\text{N}·{\text{m}}^{2}}\right)\left(4\text{π}\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}{10}^{-7}\frac{\text{T}·\text{m}}{\text{A}}\right)}}=3.00\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}{10}^{8}\phantom{\rule{0.2em}{0ex}}\text{m/s},$

which is the speed of light . Imagine the excitement that Maxwell must have felt when he discovered this equation! He had found a fundamental connection between two seemingly unrelated phenomena: electromagnetic fields and light.

Check Your Understanding The wave equation was obtained by (1) finding the E field produced by the changing B field, (2) finding the B field produced by the changing E field, and combining the two results. Which of Maxwell’s equations was the basis of step (1) and which of step (2)?

(1) Faraday’s law, (2) the Ampère-Maxwell law

So far, we have seen that the rates of change of different components of the E and B fields are related, that the electromagnetic wave is transverse, and that the wave propagates at speed c . We next show what Maxwell’s equations imply about the ratio of the E and B field magnitudes and the relative directions of the E and B fields.

We now consider solutions to [link] in the form of plane waves for the electric field:

${E}_{y}\left(x,t\right)={E}_{0}\phantom{\rule{0.2em}{0ex}}\text{cos}\phantom{\rule{0.2em}{0ex}}\left(kx-\omega t\right).$

We have arbitrarily taken the wave to be traveling in the +x -direction and chosen its phase so that the maximum field strength occurs at the origin at time $t=0$ . We are justified in considering only sines and cosines in this way, and generalizing the results, because Fourier’s theorem implies we can express any wave, including even square step functions, as a superposition of sines and cosines.

At any one specific point in space, the E field oscillates sinusoidally at angular frequency $\omega$ between $+{E}_{0}$ and $\text{−}{E}_{0},$ and similarly, the B field oscillates between $+{B}_{0}$ and $\text{−}{B}_{0}.$ The amplitude of the wave is the maximum value of ${E}_{y}\left(x,t\right).$ The period of oscillation T is the time required for a complete oscillation. The frequency f is the number of complete oscillations per unit of time, and is related to the angular frequency $\omega$ by $\omega =2\pi f$ . The wavelength $\lambda$ is the distance covered by one complete cycle of the wave, and the wavenumber k is the number of wavelengths that fit into a distance of $2\text{π}$ in the units being used. These quantities are related in the same way as for a mechanical wave:

$\omega =2\pi f,\phantom{\rule{1.2em}{0ex}}f=\frac{1}{T},\phantom{\rule{1.2em}{0ex}}k=\frac{2\pi }{\lambda },\phantom{\rule{1.2em}{0ex}}\text{and}\phantom{\rule{1.2em}{0ex}}c=f\lambda =\omega \text{/}k.$

Given that the solution of ${E}_{y}$ has the form shown in [link] , we need to determine the B field that accompanies it. From [link] , the magnetic field component ${B}_{z}$ must obey

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