Chapter 33

Electromagnetic WavesToday’s information age is based almost entirely on the physics ofToday s information age is based almost entirely on the physics of electromagnetic waves. The connection between electric and magnetic fields to produce light is one of the greatest achievements produced by physics, and electromagnetic waves are at the core of many fields in science andand electromagnetic waves are at the core of many fields in science and engineering.

In this chapter we introduce fundamental concepts and explore the properties of electromagnetic waves.

(33-1)

The wavelength/frequency range in which electromagnetic (EM) waves (light)

Maxwell’s RainbowThe wavelength/frequency range in which electromagnetic (EM) waves (light) are visible is only a tiny fraction of the entire electromagnetic spectrum.

Fig 33-2Fig. 33 2

Fig. 33-1 (33-2)

The Traveling Electromagnetic (EM) Wave, Qualitatively

An LC oscillator causes currents to flow sinusoidally, which in turn produces oscillating electric and magnetic fields, which then propagate through space as EM waves.

Next slide

Fig. 33-3gOscillation Frequency:

1ω =LC

ω

(33-3)

The Traveling Electromagnetic (EM) Wave, Qualitatively

EM fields at P looking back toward LC oscillator

1. Electric and magnetic fields are always perpendicular to direction in which wave

E Br r

perpendicular to direction in which wave is traveling transverse wave (Ch. 16).

2. is always perpendicular to . E B

→r r

y p p

3. always gives direcE B×r r

tion of wave travel.

4. and vary sinusoidally (in time and space)E Br r

y y ( p ) and are (in step) with each other.in phase

Fig. 33-4 (33-4)

Mathematical Description of Traveling EM Waves

Electric Field: ( )sinmE E kx tω= −Wave Speed:

0 0

1cμ ε

=

Magnetic Field: ( )sinmB B kx tω= −0 0μ

2π

All EM waves travel a c in vacuum

Wavenumber:2k πλ

=

Angular frequency:2πω =

EM Wave SimulationAngular frequency: ω

τ

Vacuum Permittivity: 0ε

Fig 33-5Vacuum Permeability: 0μ

Fig. 33 5

Amplitude Ratio: m

m

E cB

= Magnitude Ratio:( )( )

E tc

B t=

(33-5)

A Most Curious Wave• Unlike all the waves discussed in Chs. 16 and 17, EM waves require no

medium through/along which to travel. EM waves can travel through empty space (vacuum)!

• Speed of light is independent of speed of observer! You could be heading toward a light beam at the speed of light, but you would still measure c as the speed of the beam!p

299 792 458 m/sc =

(33-6)

The Traveling EM Wave, Quantitatively

Changing magnetic fields produce electric fields, Faraday’s law of induction:

Induced Electric Field

Φd

hdEhEdEEsdE =−+=⋅∫ ])[(rr∫

Φ−=⋅

dtdsdE Brr

( ) ( )B B h dxΦ =dB dE dBh d h d

hdEhEdEEsdE =+=⋅∫ ])[(

dB dE dBh dE h dxdt dx dt

→ = → = −

E B∂ ∂= −

Fig. 33-6

( ) ( )cos and cosm mE BkE kx t B kx tx t

ω ω ω∂ ∂= − = − −

∂ ∂

x t∂ ∂

x t∂ ∂

( ) ( )cos cos mm m

m

EkE kx t B kx t cB

ω ω ω− = − − → = (33-7)

The Traveling EM Wave, Quantitatively

Changing electric fields produce magnetic fields, Maxwell’s law of induction:

Induced Magnetic Field

Φd∫

Φ=⋅

dtdsdB E

00εμrr

hdBhBdBBsdB −=−+−=⋅∫ ])[(rr

( ) ( ) EE

d dEE h dx h dxdt dtΦ

Φ = → =

dB⎛ ⎞

hdBhBdBBsdB =+=⋅∫ ])[(

0 0 dBh dB h dxdt

μ ε ⎛ ⎞→ − = ⎜ ⎟⎝ ⎠

B Eμ ε∂ ∂− =Fig. 33-7

0 0x tμ ε=

∂ ∂

( ) ( )0 0cos cosm mkB kx t E kx tω μ ε ω ω− − = − −

g

( )0 0 0 0 0 0

1 1 1m

m

E c cB k cμ ε ω μ ε μ ε

= = = → = (33-8)

Energy Transport and the Poynting Vector

EM waves carry energy. The rate of energy transport in an EM wave

is characterized by the Poynting vector, :Sr

Poynting Vector:0

1S E Bμ

= ×r r r

The magnitude of S is related to the rate at which energy is transported by a wave across a unit area at any instant (inst). The unit for S is (W/m2).

inst inst

energy/time powerarea area

S ⎛ ⎞ ⎛ ⎞= =⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

The direction of at any point gives the wave's travel directionand the direction of energy transport at that point.

Sr

gy p p

(33-9)

Energy Transport and the Poynting Vector

Instantaneous energy flow rate:

0

1S EBcμ

=Since

1 d i

E B E B EB

ES EB B

⊥ → × =r r r r

0μ

Note that S is a function of time. The time-averaged value for S, Savg is also called the intensity I of the wave

0

1 and since ES EB Bcμ

= =

called the intensity I of the wave.

avgenergy/time power

area areaI S ⎛ ⎞ ⎛ ⎞= = =⎜ ⎟ ⎜ ⎟

⎝ ⎠ ⎝ ⎠avg avgarea area⎝ ⎠ ⎝ ⎠

( )2 2 2avg avg avg

0 0

1 1 sinmI S E E kx tc c

ωμ μ

⎡ ⎤ ⎡ ⎤= = = −⎣ ⎦ ⎣ ⎦0 0μ μ

rms 2mEE =

2⎛ ⎞

2rms

0

1I Ecμ

=

( )2

222

0 0 000 0

1 1 1 12 2 2 2E B

Bu E cB B uε ε εμμ ε

⎛ ⎞⎛ ⎞= = = = =⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎝ ⎠⎝ ⎠ (33-10)

Variation of Intensity with Distance

Consider a point source S that is emitting EM waves isotropically (equally in all directions) at a rate PS. Assume that the energy of waves is conserved as they spread from source.

How does the intensity (power/area) change

ith di t ?with distance r?

2

powerarea 4

SPIrπ

= =

Fig. 33-8(33-11)

EM waves have linear momentum as well as energy→light can exert pressure.Radiation Pressure

Sr

ΔTotal absorption:

Up ΔΔ = r

Ip =incidentS pΔ

fl t dSr

p pc rp

c

incidentSr

reflectedS

pΔTotal reflectionback along path:

2 UpcΔ

Δ =2

rIp

c=

(total absorption)

U IA tIAF

Δ = Δ

=

incident

pFt

Δ=Δ (total absorption)

2 (total reflection back along path)

FcIAFc

=

tΔpower energy/timearea area

I = =

Radiation Pressurer

cFpA

= U tA

Δ Δ= (33-12)

Polarization

The polarization of light describes how the electric field in the EM wave oscillates.

Vertically planepolarized (or linearly polarized)

Fig. 33-10 (33-13)

Polarized Light

Unpolarized or randomly polarized light has its instantaneous polarization direction vary randomly with time.

One can produce unpolarized light by the addition (superposition) of twoaddition (superposition) of two perpendicularly polarized waves with randomly varying amplitudes. If the two perpendicularly polarized waves have fixedperpendicularly polarized waves have fixed amplitudes and phases, one can produce different polarizations such as circularly or elliptically polarized light.

Fig. 33-11p y p g

(33-14)

Polarizing Sheet

I0

I

Fig. 33-12

Only the electric field component along the polarizing direction of polarizing sheet is passed (transmitted); the perpendicular component is blocked (absorbed).

(33-15)

Intensity of Transmitted Polarized Light

Intensity of transmitted light,unpolarized 0

12

I I=unpolarized incident light:

02

Since only the component of the incident electric field E parallel to the polarizing axis is transmitted.

cosE E E θFig. 33-13

transmitted cosyE E E θ= =

Intensity of transmitted light, 2I I θpolarized incident light:

20 cosI I θ=

For unpolarized light θ varies randomly in time:

( ) ( )2 20 0 0avg avg

1cos cos2

I I I Iθ θ= = =

For unpolarized light, θ varies randomly in time:

(33-16)

Reflection and RefractionAlthough light waves spread as they move from a source, often we can approximate its travel as being a straight line → geometrical optics.

What happens when a narrow beam of li ht t l f ?light encounters a glass surface?

Reflection: 'θ θ=Law of Reflection

Reflection: 1 1θ θ=

Snell’s Law

Refraction: 2 2 1 1sin sinn nθ θ=

1sin sinnθ θ=Fig. 33-17 2 12

sin sinn

θ θ=

n is the index of refraction of the material. (33-17)

Snell’s Law

For light going from n1 to n2:

• n2 = n1 → θ2 = θ12 1 2 1

• n2 > n1 → θ2<θ1, light bent toward normal

• n2 < n1 → θ2 > θ1, light bent away from normal

Refraction: 2 2 1 1sin sinn nθ θ=

Fig. 33-18(33-18)

Chromatic DispersionThe index of refraction n encountered by light in any medium except vacuum depends on the wavelength of the light. So if light consisting of different wavelengths enters a material, the different wavelengths will be refracted diff l h i di idifferently → chromatic dispersion.

Fig. 33-19 Fig. 33-20N2,blue>n2,red

Chromatic dispersion can be good (e.g., used to analyze wavelength composition of light) or bad (e.g., chromatic aberration in lenses).

(33-19)

Chromatic Dispersion

Chromatic dispersion can be good (e.g., used to analyze wavelength composition of light)

prism

Fig. 33-21

or bad (e.g., chromatic aberration in lenses)

lens (33-20)

Rainbows

Sunlight consists of all visible colors and water is dispersive, so when sunlight is refracted as it enters water droplets is reflected off the back surface andwater droplets, is reflected off the back surface, and again is refracted as it exits the water drops, the range of angles for the exiting ray will depend on the color of the ray. Since blue is refracted more strongly than red, only y g y , ydroplets that are closer to the rainbow center (A) will refract/reflect blue light to the observer (O). Droplets at larger angles will still refract/reflect red light to the observer.

What happens for rays that reflect twice off the back f f th d l t ?surfaces of the droplets?

Fig. 33-22(33-21)

Total Internal ReflectionFor light that travels from a medium with a larger index of refraction to a medium with a smaller index of refraction n1>n1 → θ2>θ1, as θ1 increases, θ2will reach 90o (the largest possible angle for refraction) before θ1 does.

1 2 2sin sin 90n n nθ = ° =n2

1 2 2sin sin 90cn n nθ

Critical Angle: 1 2sin nθ −=

n1

C g1

sinc nθ

When θ2> θc no light is

Fig. 33-24

refracted (Snell’s law does not have a solution!) so no light is transmitted → Total I t l R fl tiTotal internal reflection can be used, for Internal ReflectionTotal internal reflection can be used, for

example, to guide/contain light along an optical fiber. (33-22)

Polarization by ReflectionWhen the refracted ray is perpendicular to the reflected ray, the electric field y p p y,parallel to the page (plane of incidence) in the medium does not produce a reflected ray since there is no component of that field perpendicular to the reflected ray (EM waves are transverse).

Applications1. Perfect window: since parallel polarization

is not reflected, all of it is transmitted2. Polarizer: only the perpendicular component

is reflected, so one can select only this component of the incident polarization

Brewster’s Law90B rθ θ+ = °

( )1 2

1 2 2

sin sinsin sin 90 cos

B r

B B B

n nn n n

θ θθ θ θ

=

= ° − =Fig. 33-27

Brewster Angle: 1 2

1

tanBnn

θ −=In which direction does light reflecting off a lake tend to be polarized? (33-23)