# 16.2 Plane electromagnetic waves  (Page 3/5)

 Page 3 / 5
${E}_{y}\left(x,t\right)=f\left(\xi \right)\phantom{\rule{1.5em}{0ex}}\text{where}\phantom{\rule{0.2em}{0ex}}\xi =x-ct.$

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

#### Questions & Answers

relation between Celsius and Kelvin
Anish Reply
Newton's second laws is call with
Dyutee Reply
Really
Arzoodan
what is mean by thermodynamics
Prasad Reply
it is study about temperature and it's equilibrium
thiru
Its the study of heat and its relation with others kind of energy
Antonio
state caulombs law clearly
constand Reply
show mathematically that an electron has the greater speed than the proton when they attract each other
ezra Reply
show mathematically that an electron has the greater speed than the proton when they attract each other
srikanta
@ezra & srikanta; for electrons: a=ke^2/(mr^2) and for protons: a=kp^2/(mr^2)
Sikandar
what is electrostatics
Hero Reply
the study of charge at rest
Gulzar
@Hero; the study of charges at rest is the electrostatics
Sikandar
okay what is electrostatic?
Abd
charge at rest
Nawal
set of character...
Arzoodan
oky
Abd
Gauss law, electric fields, dipoles,...
Antonio
good
Abd
A proton initially at rest falls through a p.d of 25000V. what speed does it gain?
Minister Reply
@Minister; use equation v= sq root(2×eV/m)
Sikandar
what is the reaction of heat on magnet
ORIZINO Reply
Magnetization decreases with increase in temperature. But in case of diamagnetic substance heat has no role on magnetization.
srikanta
what is a physical significant of electric dipole moment .
PRANAB Reply
A dipole moment it's a mechanical electrical effect used in nature
Antonio
what is the uses of carbon brushes in generator
Malik Reply
to minimize heat
constand
at what temperature is the degree Fahrenheit equal to degree Celsius
Grace Reply
Celsius and Faharaneith are different, never equal
Antonio
find their liners express of n=a+b/T² ( plot graph n against T)
Donsmart Reply
Radio Stations often advertis "instant news,,if that meens you can hear the news the instant the radio announcer speaks it is the claim true? what approximate time interval is required for a message to travel from Cairo to Aswan by radio waves (500km) (Assume the waves Casbe detected at this range )
mahmod Reply
what is growth and decay
Pawan Reply
Can someone please predict the trajectory of a point charge in a uniform electric field????
erlinda Reply

### Read also:

#### Get the best University physics vol... course in your pocket!

Source:  OpenStax, University physics volume 2. OpenStax CNX. Oct 06, 2016 Download for free at http://cnx.org/content/col12074/1.3
Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

Would you like to follow the 'University physics volume 2' conversation and receive update notifications?

 By Qqq Qqq By By By