Chapter 2Wave Motion

Lecture 4

Plane waves 3D Differential wave equation Spherical waves Cylindrical waves

3-D waves: plane waves(simplest 3-D waves)

All the surfaces of constant phase of disturbance form parallel planes that are perpendicular to the propagation direction

3-D waves: plane waves(simplest 3-D waves)

All the surfaces of constant phase of disturbance form parallel planes that are perpendicular to the propagation direction

Unit vectors

An equation of plane that is perpendicular to kji ˆˆˆ

zyx kkkk

aconstrk

All possible coordinates of vector r are on a plane k

Can construct a set of planes over which varies in space harmonically:

rkAr sin

rkAr cosor

rkiAer or

Plane waves

rkr sin The spatially repetitive nature

can be expressed as:

kkrr

In exponential form:

kirkikkrkirki eAeAeAer /

For that to be true: 12 ie

2k

2

k

Vector k is called propagation vector

Plane waves: equation

rkiAer This is snap-shot in time, no time dependence

To make it move need to add time dependence the same way as for one-dimensional wave:

trkiAetr , Plane wave equation

Plane wave: propagation velocity

Can simplify to 1-D case assuming that wave propagates along x:

trkiAetr ,

i||r tkxiAetr ,

We have shown that for 1-D wave phase velocity is:

k

vThat is true for any direction of k+ propagate with k- propagate opposite to k

More general case: see page 26

Example: two plane wavesSame wavelength: k1= k2=k=2/,Write equations for both waves.Solution:

Same speed v:1=2==kv

trkiAe

Dot product:zkykxkrk zyx

Wave 1: zkrk 11

tzkieA 111

directionWave 2: zkykrk cossin 222

tzykieA cossin22

2

tkzA cos11 tzykA cossincos22

Note: in overlapping region = 1 + 2

Plane waves: Cartesian coordinates

trkiAetr , zkykxkrk zyx

tzyxkiAetzyx ,,, , , - direction cosines of k

tzkykxki zyxAetzyx ,,,

Wave eq-ns in Cartesian coordinates:

222zyx kkkk

1222

Importance of plane waves:• easy to generate using any harmonic generator• any 3D wave can be expressed as superposition of plane waves

Three dimensional differential wave equationTaking second derivatives for tzyxkiAetzyx ,,,can derive the following:

222

2

kx

222

2

ky

222

2

kz

+

+

22

2

2

2

2

2

kzyx

22

2

t

2

2

21

t

combine and use: vk

3-D differential wave equation

2

2

22

2

2

2

2

2 1tzyx

v

Three dimensional differential wave equation

2

2

22

2

2

2

2

2 1tzyx

vAlternative expression

Use Laplacian operator:

2

2

2

2

2

22

zyx

2

2

22 1

t

v

Using =kv, we can rewrite tzyxkiAetzyx ,,, tzyxikAetzyx v ,,,function of tzyx v

tzyxftzyx v ,,,It can be shown, that:

tzyxgtzyx v ,,,

f, g are plane-wave solutions of the diff. eq-n, provided that are twice differentiable.Not necessarily harmonic!

In more general form, the combination is also a solution: tkkrgCtkkrfCtzyx vv //,,, 21

ExampleGiven expression: , where a>0, b>0 2, cbtaxtx

Does it correspond to a traveling wave? What is its speed?

Solution:1. Function must be twice differentiable

acbtaxx

2

22

2

2ax

bcbtaxt

2

22

2

2bt

2

2

22

2

2

2

2

2 1tzyx

v

2. Speed:

22

2 212 bav

ab

v

Direction: negative x direction

ExampleGiven expression ,where a>0, b>0: btaxtx 2,

Does it correspond to a traveling wave? What is its speed?

Solution:1. Function must be twice differentiable

32 x

x

42

2

6 ax

x

bt

02

2

t

2

2

22

2

2

2

2

2 1tzyx

v

2. Wave equation:

06 4 ax Is not solution of wave equation!This is not a wave traveling at constant speed!

Spherical waves

2-D concentric water wavesSpherical waves originate from a point source and propagate at constant speed in all directions: waveforms are concentric spheres. Isotropic source - generates waves in all directions.

spherical waveSymmetry: introduce spherical coordinates

cossinsincossin

rzryrx

Symmetry: the phase of wave should only depend on r, not on angles:

rrr ,,

Spherical waves

cossinsincossin

rzryrx

2

2

22

2

22

2

sin1

sinsin1

1

r

r

rr

rr

2

2

22 1

t

v

r

rrr

22

2 1

Since depends only on r:

rrr 2

22 1

evaluates to the same

2

2

22

2 11t

rrr

v

Wave equation:

×r

rt

rr 2

2

22

2 1

v

Spherical waves

rt

rr 2

2

22

2 1

v

This is just 1-D wave equationIn analogy, the solution is:

trftrr v,

r

trftr v, - propagates outwards (diverging)

+ propagates inward (converging)

Note: solution blows up at r=0

In general, superposition works too:

r

trgCr

trfCtr vv

21,

Harmonic Spherical waves

r

trftr v,

Harmonic spherical wave

trkr

tr vcos, A

In analogy with 1D wave:

triker

tr vA, - source strengthA

Constant phase at any given time: kr=constAmplitude decreases with r A

Single propagatingpulse

Spherical harmonic waves

trkr

tr vcos, A

Decreasing amplitude makes sense:

Waves can transport energy (even though matter does not move)

The area over which the energy is distributed as wave moves outwards increases

Amplitude of the wave must drop!

Note: spherical waves far from source approach plane waves:

Cylindrical wavesWavefronts form concentric cylinders of infinite length

zzryrx

sincos

2

2

2

2

2

2

1

1

zr

rr

rr

Symmetry: work in cylindrical coordinates rzrr ,,

2

2

211

trr

rr

v

It similar to Bessel’s eq-n.At larger r the solution can be approximated:

Harmonic cylindrical wave

trkr

tr vcos, A

triker

tr vA,

Cylindrical waves

Harmonic cylindrical wave

trkr

tr vcos, A

triker

tr vA,

Can create a long wave source by cutting a slit and directing plane waves at it:emerging waves would be cylindrical.

Chapter 3

Electromagnetic theory, Photons.and Light

Lecture 5

Basic laws of electromagnetic theory Maxwell’s equations Electromagnetic waves Polarization of EM waves Energy and momentum

Basic laws of electromagnetic theory

221

041

rQQ

FF

Q1 Q2

F F

Coulomb force law:

Black box

rrQE

QEF

ˆ4

122

0

1

Interaction occurs via electric fieldElectric field can exist even when charge disappears (annihilation in black box)

Electric field

electric permittivity of free space

Basic laws of electromagnetic theory

Magnetic field Moving charges create magnetic field

20 ˆ

4 rrvqB

The Biot-Savart law for

moving charge

Magnetic field interacts with moving charges: BvqFmagnetic

Charges interact with both fields:

BvqEqF

(Lorentz force)

permeability of free space

Basic laws of electromagnetic theoryGauss’s Law: electric

Karl Friedrich Gauss (1777-1855)Electric field flux from an enclosed volume is proportional to the amount of charge inside

qE0

1

qSdES

0

1

If there are no charges (no sources of E field), the flux is zero: 0S SdE

More general form:

VS

dVSdE 0

1

Charge density

Basic laws of electromagnetic theoryGauss’s Law: magnetic

Magnetic field flux from an enclosed volume is zero (no magnetic monopoles)

0M

0S SdB

Basic laws of electromagnetic theoryFaraday’s Induction Law1822: Michael Faraday Changing magnetic field can result in variable electric field

dtdemf M

AC

SdBdtdldE

Formalversion

dAnSd ˆ

dAnBSdBd M ˆ

normal to area

cosBdAd M

Changing current in the solenoid produces changing magnetic field B. Changing magnetic field flux creates electric field in the outer wire.

area

angle between B and normal to the area dA

Basic laws of electromagnetic theoryAmpère’s Circuital Law

1826: (Memoir on the Mathematical Theory of Electrodynamic Phenomena, Uniquely Deduced from Experience)

All the currents in the universe contribute to Bbut only ones inside the path result in nonzero path integral

A wire with current creates magnetic field around it

Ampere’s law

pathinsideCIldB _0

AC

SdJldB

0

Current density

Incomplete!

Basic laws of electromagnetic theoryAmpère’s-Maxwell’s Law

Maxwell considered all known laws and noticed asymmetry:

AC

SdBdtdldE

0S SdB

qSdES

0

1

AC

SdJldB

0

Gauss’s

Gauss’s

Faraday’s

Ampère’s

Changing magnetic field leads to changing electric field

No similar term here

Hypothesis: changing electric field leads to variable magnetic field

Basic laws of electromagnetic theoryAmpère’s-Maxwell’s Law

AC

SdJldB

0

Ampère’s law

iSdJldBAC 00

1

02

0 ACSdJldB

The B will depend on area:

Workaround: Include term that takes into account changing electric field flux in area A2:

AC

SdtEJldB

00 Ampère’s-Maxwell’s Law:

displacement current density

Maxwell equations

AC

SdBdtdldE

0S SdB

qSdES

0

1

Gauss’s

Gauss’s

Faraday’s

Ampère-Maxwell’s

AC

SdtEJldB

00

+Lorentz force: BvqEqF

fields are defined through interaction with charges

Inside the media electric and magnetic fields are scaled. To account for that the free space permittivity 0 and 0 are replaced by and :

0 EKdielectric constant, KE>1

0 MKrelative permeability

In vacuum(free space)

Maxwell equations

AC

SdBdtdldE

0S SdB

qSdES

1

Gauss’s

Gauss’s

Faraday’s

Ampère-Maxwell’s

AC

SdtEJldB

Lorentz force: BvqEqF

+

fields are defined through interaction with charges

In matter

Maxwell equations: free space, no chargesCurrent J and charge are zero

AC

SddtBdldE

0S SdB

0S SdE

AC

SdtEldB

00

There is remarkable symmetry between electric and magnetic fields!

Integral form of Maxwell equations in free space:

no magnetic ‘charges’

no electric charges

changing magnetic field creates changing electric fieldchanging electric field creates changing magnetic field

Maxwell equations: differential form(free space)

0 E

0 B

tBE

tEB

00

Notation: kz

jy

ix

ˆˆˆ

2

2

2

2

2

22

zyx

Laplacian:

0)(

z

Ey

Ex

EEdivE zyx

ky

Ex

Ej

xE

zEi

zE

yEE xyzxyz ˆˆˆ

tB

zE

yE xyz

tB

xE

zE yzx

tB

yE

xE zxy

)(EcurlE

Electromagnetic waves

(free space)

tBE

tEB

00

Changing E field creates B fieldChanging B field creates E field

Is it possible to create self-sustaining EM field?

Can manipulate mathematically into:

2

2

002

tEE

2

2

002

tBB

Electromagnetic waves

2

2

002

tEE

2

2

002

tBB

2

2

2

2

2

22

ˆˆˆ

zyx

kx

jx

ix

2

2

002

2

2

2

2

2

2

2

002

2

2

2

2

2

2

2

002

2

2

2

2

2

tE

zE

yE

xE

tE

zE

yE

xE

tE

zE

yE

xE

zzzz

yyyy

xxxx

2

2

002

2

2

2

2

2

2

2

002

2

2

2

2

2

2

2

002

2

2

2

2

2

tB

zB

yB

xB

tB

zB

yB

xB

tB

zB

yB

xB

zzzz

yyyy

xxxx

Resembles wave equation: 2

2

22

2

2

2

2

22 1

tzyx

v

Each component of the EM field obeys the scalar wave equation, provided that

00

1

v

Light - electromagnetic wave?

00

1

vMaxwell in ~1865 found that EM wave must move at speed

At that time permittivity 0 and permeability 0 were known from electric/magnetic force measurements and Maxwell calculated

km/s 740,3101

00

v

Speed of light was also measured by Fizeau in 1949: 315,300 km/s

Maxwell wrote: This velocity is so nearly that of light, that it seems we have strong reason to conclude that light itself (including radiant heat, and other radiations if any) is an electromagnetic disturbance in the form of waves propagated through the electromagnetic field according to electromagnetic laws.

Exact value of speed of light: c = 2.997 924 58 × 108 m/sceler (lat. - fast)

Electromagnetic waveAssume: reference frame is chosen so that E=(Ex,0,0)

longitudinal wave, propagates along x

0 E

0 B

tBE

tEB

00

0

xEx Ex does not vary with x

This cannot be a wave!

Conclusion: it must be transverse wave, i.e. Ex=0. Similarly Bx=0.

Since E is perpendicular to x, we must specify its direction as a function of time

Direction of vector E in EM wave is called polarization

Simple case: polarization is fixed, i.e. direction of E does not change

0

zE

yE

xE zyx

Polarized electromagnetic waveWe are free to chose y-axis so that E field propagating along x is polarized along y: (0, Ey ,0).

0 E

0 B

tBE

tEB

00

tB

zE

yE xyz

tB

xE

zE yzx

tB

yE

xE zxy

tB

xE zy

Also: Bx=By=const (=0)

E-field of wave has only y componentB-field of wave has only z component(for polarized wave propagating along x)

In free space, the plane EM wave is transverse

Harmonic polarized electromagnetic wave

cxtEtxE yy /cos, 0

Harmonic functions are solution for wave equation:

polarized along y axis propagates along x axis

tB

xE zy

dtx

EB y

z

Find B:

cxtEc

txB yz /cos1, 0

zy cBE

This is true for any wave:- amplitude ratio is c- E and B are in-phase

Harmonic polarized electromagnetic wave

* direction of propagation is in the direction of cross-product:

BE

* EM field does not ‘move’ in space, only disturbance does.Changing E field creates changing B field and vice versa

Electromagnetic waves

Energy of EM wave

It was shown (in Phys 272) that field energy densities are:

20

2EuE

2

021 BuB

Since E=cB and c=(00)-1/2:

BE uu

- the energy in EM wave is shared equally between electric and magnetic fields

Total energy: 2

0

20

1 BEuuu BE (W/m2)

The Poynting vector

EM field contains energy that propagates through space at speed cEnergy transported through area A in time t: uAct

EBEBcBEcEcuctAtuAcS

00

000

20

11

Energy S transported by a wave through unit area in unit time:

E c2

The Poynting vector:

BES

0

1

power flow per unit area for a wave, direction of propagation is direction of S.

(units: W/m2)

John Henry Poynting (1852-1914)

The Poynting vector: polarized harmonic wave

BES

0

1

Polarized EM wave:

trkEE

cos0

trkBB

cos0

Poynting vector:

trkBES

200

0

cos1

This is instantaneous value: S is oscillating

Light field oscillates at ~10 15 Hz -most detectors will see average value of S.

Irradiance

trkBES

200

0

cos1Average value for periodic function: need to average one period only.

It can be shown that average of cos2 is: 21cos2 T

t20

000

0 221 EcBES

T

And average power flow per unit time:

Irradiance:20

0

2EcSI

T

Alternative eq-ns:

TTBcEcI 2

0

20

Usually mostly E-field component interacts with matter, and we will refer to E as optical field and use energy eq-ns with E

Irradiance is proportional to the square of the amplitude of the E field

For linear isotropic dielectric:

TEI 2v

Optical power radiant flux total power falling on some area (Watts)

Spherical wave: inverse square lawSpherical waves are produced by point sources. As you move away from the source light intensity drops

trkr

tr vcos, A

Spherical wave eq-n:

trkr

EE

cos0 trk

rBB

cos0

trkr

Br

ES

200

0

cos1

202

0 12

Er

cSIT

Inverse square law: the irradiance from a point source drops as 1/r2

Classical EM waves versus photonsThe energy of a single light photon is E=h

The Planck’s constant h = 6.626×10-34 JsVisible light wavelength is ~ 0.5 m J 104 19

1

chhE

Example: laser pointer output power is ~ 1 mWnumber of photons emitted every second:

photons/s 105.2J/photon 104J/s10 15

19

3

1

EP

Conclusion: in many every day situations the quantum nature of light is not pronounced and light could be treated as a classical EM wave