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It is left as a mathematical exercise to show, using the chain rule for differentiation, that [link] and [link] imply
The speed of the electromagnetic wave in free space is therefore given in terms of the permeability and the permittivity of free space by
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
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:
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 $\text{\u2212}{E}_{0},$ and similarly, the B field oscillates between $+{B}_{0}$ and $\text{\u2212}{B}_{0}.$ 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 $2\text{\pi}$ in the units being used. These quantities are related in the same way as for a mechanical wave:
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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