# 16.2 Plane electromagnetic waves  (Page 4/5)

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$\begin{array}{c}\frac{\partial {B}_{Z}}{\partial t}=-\frac{\partial {E}_{y}}{\partial x}\hfill \\ \frac{\partial {B}_{Z}}{\partial t}=-\frac{\partial }{\partial x}\phantom{\rule{0.2em}{0ex}}{E}_{0}\phantom{\rule{0.2em}{0ex}}\text{cos}\left(kx-\omega t\right)=k{E}_{0}\phantom{\rule{0.2em}{0ex}}\text{sin}\left(kx-\omega t\right).\hfill \end{array}$

Because the solution for the B -field pattern of the wave propagates in the + x -direction at the same speed c as the E- field pattern, it must be a function of $k\left(x-ct\right)=kx-\omega t$ . Thus, we conclude from [link] that ${B}_{z}$ is

${B}_{z}\left(x,t\right)=\frac{k}{\omega }{E}_{0}\phantom{\rule{0.2em}{0ex}}\text{cos}\phantom{\rule{0.2em}{0ex}}\left(kx-\omega t\right)=\frac{1}{c}{E}_{0}\phantom{\rule{0.2em}{0ex}}\text{cos}\phantom{\rule{0.2em}{0ex}}\left(kx-\omega t\right).$

These results may be written as

$\begin{array}{c}{E}_{y}\left(x,t\right)={E}_{0}\phantom{\rule{0.2em}{0ex}}\text{cos}\phantom{\rule{0.2em}{0ex}}\left(kx-\omega t\right)\hfill \\ {B}_{z}\left(x,t\right)={B}_{0}\phantom{\rule{0.2em}{0ex}}\text{cos}\phantom{\rule{0.2em}{0ex}}\left(kx-\omega t\right)\hfill \end{array}$
$\frac{{E}_{y}}{{B}_{z}}=\frac{{E}_{0}}{{B}_{0}}=c.$

Therefore, the peaks of the E and B fields coincide, as do the troughs of the wave, and at each point, the E and B fields are in the same ratio equal to the speed of light c . The plane wave has the form shown in [link] .

## Calculating B -field strength in an electromagnetic wave

What is the maximum strength of the B field in an electromagnetic wave that has a maximum E -field strength of 1000 V/m?

## Strategy

To find the B -field strength, we rearrange [link] to solve for B , yielding

$B=\frac{E}{c}.$

## Solution

We are given E , and c is the speed of light. Entering these into the expression for B yields

$B=\frac{1000\phantom{\rule{0.2em}{0ex}}\text{V/m}}{3.00\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}{10}^{8}\phantom{\rule{0.2em}{0ex}}\text{m/s}}=3.33\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}{10}^{-6}\phantom{\rule{0.2em}{0ex}}\text{T}.$

## Significance

The B -field strength is less than a tenth of Earth’s admittedly weak magnetic field. This means that a relatively strong electric field of 1000 V/m is accompanied by a relatively weak magnetic field.

Changing electric fields create relatively weak magnetic fields. The combined electric and magnetic fields can be detected in electromagnetic waves, however, by taking advantage of the phenomenon of resonance, as Hertz did. A system with the same natural frequency as the electromagnetic wave can be made to oscillate. All radio and TV receivers use this principle to pick up and then amplify weak electromagnetic waves, while rejecting all others not at their resonant frequency.

Check Your Understanding What conclusions did our analysis of Maxwell’s equations lead to about these properties of a plane electromagnetic wave:
(a) the relative directions of wave propagation, of the E field, and of B field,
(b) the speed of travel of the wave and how the speed depends on frequency, and
(c) the relative magnitudes of the E and B fields.

a. The directions of wave propagation, of the E field, and of B field are all mutually perpendicular. b. The speed of the electromagnetic wave is the speed of light $c=1\text{/}\sqrt{{\epsilon }_{0}{\mu }_{0}}$ independent of frequency. c. The ratio of electric and magnetic field amplitudes is $E\text{/}B=c.$

## Production and detection of electromagnetic waves

A steady electric current produces a magnetic field that is constant in time and which does not propagate as a wave. Accelerating charges, however, produce electromagnetic waves. An electric charge oscillating up and down, or an alternating current or flow of charge in a conductor, emit radiation at the frequencies of their oscillations. The electromagnetic field of a dipole antenna is shown in [link] . The positive and negative charges on the two conductors are made to reverse at the desired frequency by the output of a transmitter as the power source. The continually changing current accelerates charge in the antenna, and this results in an oscillating electric field a distance away from the antenna. The changing electric fields produce changing magnetic fields that in turn produce changing electric fields, which thereby propagate as electromagnetic waves. The frequency of this radiation is the same as the frequency of the ac source that is accelerating the electrons in the antenna. The two conducting elements of the dipole antenna are commonly straight wires. The total length of the two wires is typically about one-half of the desired wavelength (hence, the alternative name half-wave antenna ), because this allows standing waves to be set up and enhances the effectiveness of the radiation.

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