1

Chapter 33

Today’s information age is based almost entirely on the physics of electromagnetic waves. The connection between electric and magnetic fields to produce light is own of the greatest achievements produced by physics, and 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.

Electromagnetic Waves

33-

2Fig. 33-1

The wavelength/frequency range in which electromagnetic (EM) waves (light) are visible is only a tiny fraction of the entire electromagnetic spectrum

Maxwell’s Rainbow

33-

Fig. 33-2

3

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

33-

Fig. 33-3Oscillation Frequency:

1LC

ω =

Next slide

The Travelling Electromagnetic (EM) Wave, Qualitatively

4

EM fields at P looking back toward LC oscillator

33-Fig. 33-4

The Travelling Electromagnetic (EM) Wave, Qualitatively

1. Electric and magnetic fields always perpendicular to direction in which wave is travelling transverse wave (Ch. 16)

2. always perpendicular to

3. always gives direction of

E B

E BE B

→

×

r r

r r

r rwave travel

4. and vary sinusoidally (in time and space) and are (in step) with each otherE B

in phase

r r

533-

Fig. 33-5

Mathematical Description of Travelling EM Waves

Electric Field: ( )sinmE E kx tω= −

Magnetic Field: ( )sinmB B kx tω= −

Wave Speed:0 0

1cµ ε

=

Wavenumber:2k πλ

=

Angular frequency:2πωτ

=

Vacuum Permittivity: 0ε

Vacuum Permeability: 0µ

All EM waves travel a c in vacuum

Amplitude Ratio: m

m

E cB

= Magnitude Ratio:( )( )E t

cB t

=

EM Wave Simulation

6

• 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!

A Most Curious Wave

33-

299 792 458 m/sc =

7

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

33-

Fig. 33-6

The Travelling EM Wave, QuantitativelyInduced Electric Field

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

ω ω ω∂ ∂= − = − −

∂ ∂

BdE d sdtΦ

= −∫rr

gÑ( )E d s E dE Eh h dE= + − =∫

rrgÑ( ) ( )B B h dxΦ =

dB dE dBh dE h dxdt dx dt

→ = → = −

E Bx t

∂ ∂= −

∂ ∂

( ) ( )cos cos mm m

m

EkE kx t B kx t cB

ω ω ω− = − − → =

8

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

33-

The Travelling EM Wave, QuantitativelyInduced Magnetic Field

0 0EdB d s

dtµ ε Φ

=∫rr

gÑ( )B d s B dB Bh h dB= − + − =∫

ur rgÑ( ) ( ) E

Ed dEE h dx h dxdt dtΦ

Φ = → =

0 0 dBh dB h dxdt

µ ε → − =

0 0B Ex t

µ ε∂ ∂− =∂ ∂

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

Fig. 33-7

( )0 0 0 0 0 0

1 1 1m

m

E c cB k cµ ε ω µ ε µ ε

= = = → =

9

Energy Transport and the Poynting Vector

33-

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

inst inst

energy/time powerarea area

S = =

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)

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

Sr

10

Energy Transport and the Poynting Vector

33-

Instantaneous energy flow rate:

0

1S EBcµ

=

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

Since

1 and since

E B E B EB

ES EB Bcµ

⊥ → × =

= =

r r r r

avgavg avg

energy/time powerarea area

I S = = =

( )2 2 2avg avg avg

0 0

1 1 sinmI S E E kx tc c

ωµ µ

= = = −

2m

rmsEE =

( )2

222

0 0 000 0

1 1 1 12 2 2 2E B

Bu E cB B uε ε εµµ ε

= = = = =

2

0

1rmsI E

cµ=

11

Variation of Intensity with Distance

33-Fig. 33-8

2

powerarea 4

SPIrπ

= =

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

How does the intesnity (power/area) change with distance r?

12

EM waves have linear momentum as well as energy→light can exert pressureRadiation Pressure

33-

incidentSr

p∆

incidentSr

reflectedSr

p∆

Total absorption:Upc∆

∆ =

Total reflectionBack along path:

2 Upc∆

∆ =

pFt

∆=∆

power energy/timearea area

I

U tA

= =

∆ ∆=

(total absorption)

2 (total reflection back along path)

Radiation Pressurer

U IA tIAFcIAFcFpA

∆ = ∆

=

=

=

2 r

Ipc

=

rIpc

=

13

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

Vertically plane-polarized (or linearly polarized)

Polarization

33-Fig. 33-10

14

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

Polarized Light

33-

Fig. 33-11

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

Polarized Light Simulation

15

Polarizing Sheet

33-

Fig. 33-12

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

I0

I

16

Intensity of Transmitted Polarized Light

33-

Fig. 33-13

Intensity of transmitted light,unpolarizedincident light:

012

I I=

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

transmitted cosyE E E θ= =

Intensity of transmitted light,polarized incident light:

20 cosI I θ=

( ) ( )2 20 0 0avg avg

1cos cos2

I I I Iθ θ= = =

For unpolarized light, θ varies randomly in time

17

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

Reflection and Refraction

33-

Fig. 33-17

What happens when a narrow beam of light encounters a glass surface?

Reflection: 1 1'θ θ=

Refraction: 2 2 1 1sin sinn nθ θ=

Law of Reflection

Snell’s Law

12 1

2

sin sinnn

θ θ=

n is the index of refraction of the material

18

For light going from n1 to n2

• n2 = n1 → θ2 = θ1

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

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

Sound Waves

33-Fig. 33-18

19

The 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 differently → chromatic dispersion

Chromatic Dispersion

33-

Fig. 33-19 Fig. 33-20n2blue>n2red

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

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Chromatic Dispersion

33-

Fig. 33-21

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

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

prism

lens

21

Rainbows

33-

Fig. 33-22

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, 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 droplets that are closer the 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 surfaces of the droplets?

22

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

Total Internal Reflection

33-

1 2 2sin sin 90cn n nθ = ° =

Fig. 33-24

n1

n2

Critical Angle: 1 2

1

sincnn

θ −=

When θ2> θc no light is refracted (Snell’s Law does not have a solution!) so no light is transmitted → Total Internal ReflectionTotal internal reflection can be used, for

example, to guide/contain light along an optical fiber

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Polarization by Reflection

33-

Fig. 33-27

Brewster’s Law

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

( )1 2

1 2 2

90sin sinsin sin 90 cos

B r

B r

B B B

n nn n n

θ θθ θθ θ θ

+ = °=

= ° − =

Brewster Angle: 1 2

1

tanBnn

θ −=In which direction does light reflecting off a lake tend to be polarized?

When the refracted ray is perpendicular to the reflected ray, the electric field 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).