lecture 17: electromagnetic waves -2
TRANSCRIPT
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.
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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)
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Ampere’s law and the simple plane wave• The simple plane wave must satisfy Ampere’s law in a vacuum.
• ∮" # $% = '()( *+,*- , ℎ010 ∮" # $% = "2,
• In a time dt, the electric flux through the rectangle in the xz-plane increases by an amount dΦE.
• This increase equals the flux through the shaded rectangle with area ac dt; that is, dΦE = Eac dt.
• Thus dΦE/dt = Eac.
• For consistency with the previous equation: '()(3 = 45 , 3
6 = 47898
.
We can relate the change of " across the wave front and the change of ; with time: (<=*> = '()( (<?*- , or @=@> =
45A@?@- (2). Differentiating (1) and (2) wrt. x and t and
combining the results: @A?@>A =
45A@A?@-A ,
@A=@>A =
45A@A=@-A : wave equations for E and B!
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General properties of electromagnetic waves1. From Maxwell’s equations follows that any !, " satisfy the wave equation:
∇$! ≡ &'(&)' +
&'(&+' +&
'(&,' =
./'
&'(&0' , ∇$" ≡ &'1
&)' +&'1&+' +&
'1&,' =
./'
&'1&0'
2. Maxwell’s equations imply that in an electromagnetic wave, both the electric and magnetic fields are perpendicular to the direction of propagation 23 of the wave, and to each other. 4, 5 and 26 obey the right hand rule.
3. In an electromagnetic wave, there is a definite ratio between the magnitudes of the electric and magnetic fields: E = cB.
• Unlike mechanical waves, electromagnetic waves require no medium. In fact, they travel in vacuum with a definite and unchanging speed:
• Inserting the numerical values of these constants, we obtain c = 2.997 × 108 m/s.
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Properties of electromagnetic waves• The direction of
propagation of an electromagnetic wave is the direction of the vector product of the electric and magnetic fields.
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Sinusoidal electromagnetic waves• Electromagnetic waves
produced by an oscillating point charge are an example of sinusoidal waves that are not plane waves.
• But if we restrict our observations to a relatively small region of space at a sufficiently great distance from the source, even these waves are well approximated by plane waves.
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Plane waves• When !, " do not depend on y, z, they form one dimensionnal wave (x,t)
dependence only : ∇$! ≡ &'(
&)'=
+
,'&'(
&-', ∇$" ≡
&'.
&)'=
+
,'&'.
&-'. The general
solution, when the wave travels in the positive x direction (and we have chosen the y axis to be along !) is: ! = /(1 − 34) ̂7, " = +
,/(1 − 34)89,
where f is any function!
• A wave in the negative x direction is: ! = /(1 + 34) ̂7, " = +
,/(1 + 34)(−89).
• If it is produced by a harmonic oscillator / 1 − 34 ~cos[k 1 − 34 + φ].
• 9 =$B
Cis the wave number, D is the wavelength; 93 = E =
$B
F= 2H/ is the
angular frequency, T is the period. Of course D = 3I = 3//.
• ! = !Kcos[2H()
C−
-
F) + φ] ̂7, " = "Kcos[2H(
)
C−
-
F) + φ]89, !K = 3"K
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Fields of a sinusoidal wave• Shown is a linearly polarized sinusoidal electromagnetic
wave traveling in the +x-direction.
• One wavelength of the wave is shown at time t = 0.
• The fields are shown for only a few points along the x-axis.
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Fields of a sinusoidal wave• Shown is a linearly polarized sinusoidal electromagnetic
wave traveling in the −x-direction.
• One wavelength of the wave is shown at time t = 0.
• The fields are shown for only a few points along the x-axis.
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Fields of a sinusoidal wave• We can describe electromagnetic waves by means of wave functions:
• Rules for proper planar waves traveling along x direction:
1. If x-ct: !, " and ̂$ 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.
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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.
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>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: ! = #$#% &' + &) =
*(,-. /. + 0
.1-2.) , In a wave always &' = &)!
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Energy in electromagnetic waves• The magnitude of the average value of is called the intensity.
The SI unit of intensity is 1 W/m2=m/s J/m3 .
• These rooftop solar panels are tilted to face the sun, so that they can absorb the maximum energy.
• ! = #$%&'() *'(), ! = , -. + -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
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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.
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Δ! = (35⃗$ −5⃗$)
Δ! = (0 − 5⃗$)