Lecture 17: Electromagnetic Waves -2

Lecture 17: Electromagnetic Waves -2Looking forward at …

• how and why the speed of light is related to the fundamental constants of electricity and magnetism.

• the general properties of EM waves

• how to describe the propagation of a sinusoidal electromagnetic wave.

• what determines the amount of energy and momentum carried by an electromagnetic wave.

© 2016 Pearson Education Inc.

Faraday’s law and the simple plane wave• The simple plane wave must satisfy Faraday’s law in a vacuum:

• ∮" # $% = −()*(+ , ℎ./. ∮" # $% = −"0,

• In a time dt, the magnetic flux through the rectangle in the xy-plane increases by an amount dΦB.

• This increase equals the flux through the shaded rectangle with area ac dt; that is, dΦB = Bac dt.

• Thus dΦB/dt = Bac. This and Faraday’s law imply:

• We can relate the change of " across the wave front and the change of 1with time (dx=c dt): 234(5 = 236

(+ , or 7475 =767+ (1)


Ampere’s law and the simple plane wave• The simple plane wave must satisfy Ampere’s law in a vacuum.

• ∮" # $% = '()( *+,*- , ℎ010 ∮" # $% = "2,

• In a time dt, the electric flux through the rectangle in the xz-plane increases by an amount dΦE.

• This increase equals the flux through the shaded rectangle with area ac dt; that is, dΦE = Eac dt.

• Thus dΦE/dt = Eac.

• For consistency with the previous equation: '()(3 = 45 , 3

6 = 47898

.

We can relate the change of " across the wave front and the change of ; with time: (<=*> = '()( (<?*- , or @=@> =

45A@?@- (2). Differentiating (1) and (2) wrt. x and t and

combining the results: @A?@>A =

45A@A?@-A ,

@A=@>A =

45A@A=@-A : wave equations for E and B!


General properties of electromagnetic waves1. From Maxwell’s equations follows that any !, " satisfy the wave equation:

∇$! ≡ &'(&)' +

&'(&+' +&

'(&,' =

./'

&'(&0' , ∇$" ≡ &'1

&)' +&'1&+' +&

'1&,' =

./'

&'1&0'

2. Maxwell’s equations imply that in an electromagnetic wave, both the electric and magnetic fields are perpendicular to the direction of propagation 23 of the wave, and to each other. 4, 5 and 26 obey the right hand rule.

3. In an electromagnetic wave, there is a definite ratio between the magnitudes of the electric and magnetic fields: E = cB.

• Unlike mechanical waves, electromagnetic waves require no medium. In fact, they travel in vacuum with a definite and unchanging speed:

• Inserting the numerical values of these constants, we obtain c = 2.997 × 108 m/s.


Properties of electromagnetic waves• The direction of

propagation of an electromagnetic wave is the direction of the vector product of the electric and magnetic fields.


Sinusoidal electromagnetic waves• Electromagnetic waves

produced by an oscillating point charge are an example of sinusoidal waves that are not plane waves.

• But if we restrict our observations to a relatively small region of space at a sufficiently great distance from the source, even these waves are well approximated by plane waves.


Plane waves• When !, " do not depend on y, z, they form one dimensionnal wave (x,t)

dependence only : ∇$! ≡ &'(

&)'=

+

,'&'(

&-', ∇$" ≡

&'.

&)'=

+

,'&'.

&-'. The general

solution, when the wave travels in the positive x direction (and we have chosen the y axis to be along !) is: ! = /(1 − 34) ̂7, " = +

,/(1 − 34)89,

where f is any function!

• A wave in the negative x direction is: ! = /(1 + 34) ̂7, " = +

,/(1 + 34)(−89).

• If it is produced by a harmonic oscillator / 1 − 34 ~cos[k 1 − 34 + φ].

• 9 =$B

Cis the wave number, D is the wavelength; 93 = E =

$B

F= 2H/ is the

angular frequency, T is the period. Of course D = 3I = 3//.

• ! = !Kcos[2H()

C−

-

F) + φ] ̂7, " = "Kcos[2H(

)

C−

-

F) + φ]89, !K = 3"K


Fields of a sinusoidal wave• Shown is a linearly polarized sinusoidal electromagnetic

wave traveling in the +x-direction.

• One wavelength of the wave is shown at time t = 0.

• The fields are shown for only a few points along the x-axis.


Fields of a sinusoidal wave• Shown is a linearly polarized sinusoidal electromagnetic

wave traveling in the −x-direction.

• One wavelength of the wave is shown at time t = 0.

• The fields are shown for only a few points along the x-axis.


Fields of a sinusoidal wave• We can describe electromagnetic waves by means of wave functions:

• Rules for proper planar waves traveling along x direction:

1. If x-ct: !, " and ̂$ form a right triplet of vectors (r.h. rule).

2. If x+ct: !, " and - ̂$ form a right triplet of vectors (r.h. rule).

3. !' = )"'4. The phase (the argument of the cos(phase)) is the same for E

and B. If we use sin, it is both for E and B.


Electromagnetic waves in matter• Electromagnetic waves can travel in certain types of matter, such as air, water,

or glass.

• When electromagnetic waves travel in nonconducting materials—that is, dielectrics—the speed v of the waves depends on the dielectric constant of the material.

• The ratio of the speed c in vacuum to the speed v in a material is known in optics as the index of refraction n of the material. K and Km are dependent on the frequency of the wave f. They are much smaller than their values for constant fields (f=0). Usually Km~1.


>1

Energy in electromagnetic waves• Electromagnetic waves such as those we have

described are traveling waves that transport energy from one region to another.

• The British physicist John Poynting introduced the Poynting vector, .

• The magnitude of the Poynting vector is the power per unit area in the wave, and it points in the direction of propagation.

• Flow of power per area: ! = #$#% &' + &) =

*(,-. /. + 0

.1-2.) , In a wave always &' = &)!


Energy in electromagnetic waves• The magnitude of the average value of is called the intensity.

The SI unit of intensity is 1 W/m2=m/s J/m3 .

• These rooftop solar panels are tilted to face the sun, so that they can absorb the maximum energy.

• ! = #$%&'() *'(), ! = , -. + -0 = , 1%

2 &'()2 + #

2$%*'()2 .

• In a similar way, the average power for AC Ohmic circuits is

456 =!(7(2 = !'()7'() =

7'()2

9


Electromagnetic radiation pressure• EM waves carry momentum (per unit area, per unit time) : ! = #

$ , thus exerting radiation pressure on surfaces

• %&'( = )*+,- = − Δ! 0 1

• For example, if the solar panels on an earth-orbiting satellite are perpendicular to the sunlight, and the radiation iscompletely absorbed, the

average radiation pressure is 4.7 × 10−6 N/m2.


Δ! = (35⃗$ −5⃗$)

Δ! = (0 − 5⃗$)

Lecture 17: Electromagnetic Waves -2

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