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E y ( x , t ) = f ( ξ ) where ξ = x c t .

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

1 = ε 0 μ 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 = 1 ε 0 μ 0 .

We could just as easily have assumed an electromagnetic wave with field components E z ( x , t ) and B y ( x , t ) . The same type of analysis with [link] and [link] would also show that the speed of an electromagnetic wave is c = 1 / ε 0 μ 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 = 1 ε 0 μ 0 , we find that

c = 1 ( 8.85 × 10 −12 C 2 N · m 2 ) ( 4 π × 10 −7 T · m A ) = 3.00 × 10 8 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

Got questions? Get instant answers now!

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 ( x , t ) = E 0 cos ( k x ω t ) .

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 ω between + E 0 and E 0 , and similarly, the B field oscillates between + B 0 and B 0 . The amplitude of the wave is the maximum value of E y ( x , t ) . 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 ω by ω = 2 π f . The wavelength λ 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 π in the units being used. These quantities are related in the same way as for a mechanical wave:

ω = 2 π f , f = 1 T , k = 2 π λ , and c = f λ = ω / 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

how to draw a diagram of a triode
Ssempala Reply
whate is fckg diagrame?
Arzoodan
why do we use integration?
Monalisa Reply
To know surfaces below graphs.
Jan
To find a Primitive function. Primitive function: a function that is the origin of another
playdoh
yes
Dharmdev
what is laps rate
Dharmdev
Г=-dT/dZ that is simply defination
Arzoodan
what is z
Dharmdev
to find the area under a graph or to accumulate .e.g. sum of momentum over time is no etic energy.
Naod
Z is alt.,dZ altv difference
Arzoodan
what is the Elasticty
Heeran Reply
it is the property of the by virtue of it regains it's original shape after the removal of applied force (deforming force).
Prema
property of the material
Prema
which type of cable is suitable for patrol station wiring
Eric Reply
what is calorimeter
odionyenma Reply
heat measuring device
Suvransu
What is mean electric potential
Biren Reply
Electric density formula
Biren Reply
Int E•dA
Vineet
what is relation betweeen potential energy of a system and work done in bringing last chaege from infinity
Shikhar Reply
Einstein and his general theory of relativity, a very nice concept,which revolutionize our modern world.
Antares Reply
what's the theory of relativity?
Piyali
it related to universe time period
jyotirmayee
it is a theory in which time is taken as relative,for diff. motion of objects.
Antares
no !!!
jyotirmayee
it is E=mc'square
jyotirmayee
what is black hole?
jyotirmayee
E=mc^2 is just a reln which sums up everything dude
Antares
what does e=mc^2 stand for?explain
Piyali
m not dude
jyotirmayee
e for speed of light
jyotirmayee
c for speed of light
jyotirmayee
m for mass of object
jyotirmayee
black hole is a super massive object in space gravitation pull of which s so strong dt evn light cannot escape. John wheeler 1st detected it, n concept of it ws given by Einstein himself from his general theory of relativity. Basically v cn si 3 types of it super massive, interstellar n intermediate
Antares
which topic now u studying?
jyotirmayee
e defined? sorry it,s wrong
jyotirmayee
how black holes r formed?
Piyali
e is not speed of light
jyotirmayee
black holes formed when the centre of very massive star collapsed itself
jyotirmayee
e is energy.
Antares
what do u mean by quantum mechanics?
jyotirmayee
in most of d cases black holes r formed by massive collapsed star or star system
Antares
sahoo u don't seem to understand relative physics, plz study that first.
Antares
Apollo is the name of a satellite !!!!!
jyotirmayee
quantum mechanics is d study of physics describing nature at d smallest level of energy of atoms n subatomic particles.
Antares
qm means explains about the microscopic particles..
KRANTHI
nope name of d sun god
Antares
why the mercury used in thermometers?
jyotirmayee
Ali brother ur exactly spelling is wrong,,,,,
jyotirmayee
why the colour of tube light white?
jyotirmayee
why the mouth became red colour ,,by the regular eating of leaf called ,,betel combining with areca?
jyotirmayee
all formula for calculate specific latent heat of any substance
Idowu Reply
hello
Nigar
real
Nigar
c'mon guys.. let's talk Physics.
Yoblaze
mass multiplied with latent heat of a substance
Yoblaze
you both?
jyotirmayee
in which class?
jyotirmayee
physics is the only subject which underestimated chemistry ,,bio
jyotirmayee
how capacitor is made in inst?
Piyali
what is dipole moment
Bilal Reply
it is the product of electric charges and distance between the two charges
Shikhar
mishra true thanks dia
Ssempala
product of separation of the poles, the rest shikhar got is right
Brad
brad is separation and distance ,are they different?
Ssempala
What is actually a dipole? I know charge separated by a certain distance.... but what does that really mean? what happens in a dipole? why are the charge of same magnitude?
Monalisa
Dipoles forming as a result of the unbalanced distribution of electrons in asymmetrical molecules
Heeran
What is dielectric
Ashis Reply
its a type of medium. generally poor conductors. but their conductivity can be changed
vedanth
you just have to add impurities
vedanth
Thanks
Ssempala
grt
Ssempala
a material which behave as conductor
Shikhar
insulating material, energy level for electron transfer is very high e.g used to increase a magnetic field in a capacitor
Brad
What is the difference between specific heat capacity and heat capacity? Give the equations
elly Reply
presentation on power
Dyutee Reply
relation between Celsius and Kelvin
Anish Reply
0" degree Celsius=273kelvin
jyotirmayee

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Source:  OpenStax, University physics volume 2. OpenStax CNX. Oct 06, 2016 Download for free at http://cnx.org/content/col12074/1.3
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