Chapter 3

Electromagnetic Theory, Photons,and Light

Lecture 5

Maxwell equations

AC SdBdtdldE 0S SdB qSdES 1

Gausss

Gausss

Faradays

Ampre-Maxwells

ACSd

tEJldB

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

00There 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: kzj

yi

x

2

2

2

2

2

22

zyx

Laplacian:

0)(

zE

yE

xEEdivE zyx

ky

Ex

Ej

xE

zEi

zE

yEE xyzxyz

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

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: 22

22

2

2

2

2

22 1

tzyx

v

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

00

1v

Light - electromagnetic wave?

00

1vMaxwell 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

vSpeed of light was also measured by Fizeau in 1849: 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, 0Harmonic functions are solution for wave equation:

polarized along y axis propagates along x axis

tB

xE zy

dtxEB yzFind 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

202

1 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: 20

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

1Polarized EM wave: trkEE cos0 trkBB cos0

Poynting vector:

trkBES 2000 cos1This is instantaneous value: S is oscillating

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

Irradiance

trkBES 2000 cos1Average value for periodic function: need to average one period only.It can be shown that average of cos2 is: 21cos2

Tt

20

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)

(used to be called Intensity)

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

trkr

tr vcos, ASpherical wave eq-n:

trkr

EE

cos0 trkr

BB

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

Radiation pressure

Using classical EM theory Maxwell showed that radiation pressure equals the energy density of the EM waves:

2

0

20

21

2BEu

P ucS ctSt P

This is the instantaneous pressure that would be exerted on a perfectly absorbing surface by a normally incident beam

Average pressure: cI

ctS

t TT

P (N/m2)

* for reflecting surface pressure doubles

Radiation pressure applicationStar wars episode 2

http://www.livescience.com/32593-how-do-solar-sails-work-.html

NASA to Launch World's Largest Solar Sail in November 2014: Sunjammerhttp://www.sunjammermission.com

Example problemA laser pointer emits light at 630 nm in xy plane at =450 to axis x(counter clock-wise). The light is polarized along axis z , beam cross-section is A=1 mm2 and its power is P=1 mW.1. Write an equation of E and B components of this EM wave for the region of the beam.

x

y

z trkEE cos0Find : c22 Find k: sincos2 jik Find E0:

Irradiance: 2002EcI

AP

00 2 AcPE k2 00 AcPE

tcrjiAcPE 2sincos2cosk2 0

Electric field:

B

E