08/28/2013 PHY 530 -- Lecture 01 1

Light is electromagnetic radiation!

• = Electric Field

• = Magnetic Field

• Assume linear, isotropic, homogeneous media.

),( txE

),( txB

08/28/2013 PHY 530 -- Lecture 01 2

Maxwell’s Equations

• Published by J.C. Maxwell in 1861 in the paper “On Physical Lines of Force”.

• Unite classical electricity and magnetism.• Predict the propagation of electromagnetic energy

away from time varying sources (current and charge) in the form of waves.

08/28/2013 PHY 530 -- Lecture 01 3

Maxwell’s Equations

• Four partial differential equations involving E, B that govern ALL electromagnetic phenomena.

• Gauss’s Law (elec, mag)

• Faraday’s Law, Ampere’s Law

08/28/2013 PHY 530 -- Lecture 01 4

Gauss’s Law (elec)

E

q

Charge density

Electric permittivityconstant of medium

Total chargeenclosed

Electric charges give rise to electric fields.

dS – outward normal

E dS = q

08/28/2013 PHY 530 -- Lecture 01 5

Gauss’s Law (mag)

0 B

No Magnetic Monopoles!

B dS = 0

08/28/2013 PHY 530 -- Lecture 01 6

Faraday’s Law

= mag. flux

A changing B field gives rise to an E field E field lines close on themselves (form loops)

where:

08/28/2013 PHY 530 -- Lecture 01 7

Ampere’s Law

t

E

jΒ

If E const in time:

Where:

Magneticpermeabilityof medium

Electric currents give rise to B fields.

i Electric current

j = current density

08/28/2013 PHY 530 -- Lecture 01 8

What Maxwell’s Equations Imply

In the absence of sources, all components of E, B satisfy the same (homogeneous) equation:

02

22

t

EE

02

22

t

BB

The properties of an e.m. wave (direction of propagation, velocity of propagation, wavelength, frequency) can be determined by examining the solutions to the wave equation.

08/28/2013 PHY 530 -- Lecture 01 9

What does it mean to satisfy the wave equation?

),( txImagine a disturbance traveling along thex coordinate (1-dim case).

0t 0tt ),( tx

x

08/28/2013 PHY 530 -- Lecture 01 10

What does a wave look like mathematically?

)()(),( vtxgvtxftx

General expression for waves traveling in +ve, -ve directions:

Argument affects the translation of wave shape.

is the velocity of propagation.v

08/28/2013 PHY 530 -- Lecture 01 11

Waves satisfy the wave equation

• Try it for f! Use the chain rule, differentiate:

• This is the (homogeneous) 1-dim wave equation.

01

2

2

22

2

t

f

vx

f

08/28/2013 PHY 530 -- Lecture 01 12

E, B satisfy the 3-dim wave equation!!

2

1

v

01

2

2

22

tv

can be zyxzyx BBBEEE or...,,,,,

and

08/28/2013 PHY 530 -- Lecture 01 13

Index of Refraction (1)

1

vOkay, Velocity of light in a medium dependent on medium’s electric,magnetic properties.

In free space:

m/s103.00

H/m1026.1

F/m1085.8

8

60

120

cv

08/28/2013 PHY 530 -- Lecture 01 14

Index of Refraction (2)

For any l.i.h. medium, define index of refraction as:

00

vcn

NOTE: dimensionless.

08/28/2013 PHY 530 -- Lecture 01 15

Index of Refraction (3)

medium n

vacuum 1, by definition

Air 1.0003

Water 1.33

Flint glass 1.5

Crown glass 1.7

Diamond 2.417

08/28/2013 PHY 530 -- Lecture 01 16

Plane waves

Back to the 3-dim wave equation, but assume

),( tz

z

x

y has constant valueon planes:

08/28/2013 PHY 530 -- Lecture 01 17

Seek solution to wave eqn

Solving PDEs is hard, so assume solution of the form:

)()(),( tTzZtz

(so-called “separable” solution…)

Now, Becomes:01

2

2

22

tv

08/28/2013 PHY 530 -- Lecture 01 18

Voilà! Two ordinary differential equations!

22

2

22

2

2

2

22

2

const)(

)(

1)(

)(

1

)()(

1)()(

kdt

tTd

tTvdz

zZd

zZ

dt

tTdzZ

vdz

zZdtT

022

2

Zkdz

Zd022

2

2

Tkvdt

Tdand

Note!

08/28/2013 PHY 530 -- Lecture 01 19

We know the solutions to these...

tik

tikk

ikzk

ikzkk

tT

zZ

ee)(

ee)(

where222 kv .

(Sines and cosines!)

08/28/2013 PHY 530 -- Lecture 01 20

How to build a wave

Choose w positive, +ve z dir, then have

)(1

)(0 ee),( tkzitkzitz

Any linear combination of solutions of this form isalso a solution.

Start with sines and cosines, make whatever shapelike.

08/28/2013 PHY 530 -- Lecture 01 21

Let’s get physical

Sufficient to study )(0 e tkzi

2

2

k wavelength

frequency

Harmonic wave

kz-ωt - phase (radians) ω - angular frequencyk – propagation number/vector

08/28/2013 PHY 530 -- Lecture 01 22

3-D wave equation

)(0 e),( tit xkx

Solution:

Reduces to the 1-D case when

kzkzk ˆˆ||

08/28/2013 PHY 530 -- Lecture 01 23

Back to Plane Waves

Assume we have)(

0 e),( tkzit ExE(plane waves in the z-direction, E0 a constant vector)

0

yx

EEE

Eik

z

Similar equations for B.

,

EE it

08/28/2013 PHY 530 -- Lecture 01 24

Electromagnetic Waves are Transverse

Differentiate first equation of previous slide,can show then using Maxwell’s equations that:

0zkE

0zkB Byx )ˆˆ( xy EEk

Eyx2

)ˆˆ(v

BBk xy

Try it!

08/28/2013 PHY 530 -- Lecture 01 25

EM Waves are Transverse (2)

This implies:

0Ek

0Bk BEk

EBk2v

Fields must be perpendicular to the propagation direction!

08/28/2013 PHY 530 -- Lecture 01 26

EM Waves are Transverse (3)

Also, fields are in phase in the absence of sources and E is perpendicular to B since

BEk ˆ

k

E

B

08/28/2013 PHY 530 -- Lecture 01 27

What light looks like close up

Magnetic Field Waves

Electric Field Waves

The Electric and Magneticcomponents of light are perpendicular (in vacuum).

+

Movingcharge(s)

Waves propagate withspeed 3x108 m/s.

08/28/2013 PHY 530 -- Lecture 01 28

The Poynting Vector

S is parallel to the propagation direction. In free space,

S gives us the energy transport of waveform. Energy/time/area

I = <|S|>time=1/2(c ε0) E02 - irradiance (time average of the

magnitude of the Poynting vector)

08/28/2013 PHY 530 -- Lecture 01 29

The Electromagnetic Spectrum

08/28/2013 PHY 530 -- Lecture 01 30

The Electromagnetic Spectrum