Wave SolutionsElectromagnetic Waves

•Maxwell’s Equations, no sources:•Changing E-flux creates B-field•Changing B-flux creates E-field•Can we find a self-sustaining electromagnetic solution with no sources?•Let’s try the following:

0

0

, , , sin

, , , sin

x y z t kx t

x y z t kx t

E E

B B

•I could have used cosine instead, it makes no difference•I chose arbitrarily to make it move in the x-direction•We don’t know – yet – anything about k, , E0, or B0.

0 0

0t

t

E BB E

E B

Does It Satisfy Gauss’s Laws?

0yx zEE E

x y z

•E depends only on x, so there’s only one term in this derivative:

0

0

, , , sin

, , , sin

x y z t kx t

x y z t kx t

E E

B B

•This implies E0x = 0•Same argument applies for magnetic fields

0 0xE

0 0xB

Electromagnetic waves are transverse

0 xEx

0 sinxE kx t

x

0 cosxE k kx t

0yx zBB B

x y z

•Note that the electric and magnetic fields are perpendicular to the direction the wave is traveling

Does it Satisfy Faraday’s Law?

•All terms in the first equation vanish (Bx = 0)•The others are non-trivial:

0

0

, , , sin

, , , sin

x y z t kx t

x y z t kx t

E E

B By xz

yx z

y x z

E BEy z t

BE Ez x t

E E Bx y t

yzBE

x t

0 0sin sinz yE kx t B kx tx t

0 0cos cosz yE k kx t B kx t 0 0z yE k B

0 0y zE k B •Similarly, from the third equation:

Does it Satisfy Ampere’s Laws•Very similar calculations to previous slide

0

0

, , , sin

, , , sin

x y z t kx t

x y z t kx t

E E

B B

0 0 0 0z yB k E 0 0 0 0y zB k E

0 0

0 0

0 0

y xz

yx z

y x z

B EBy z t

EB Bz x t

B B Ex y t

•Multiply each of these equations

0 0z yE k B 0 0y zE k B

2 20 0 0 0 0 0z y y zB E k E B 2 2

0 0 0 0 0 0y z z yB E k E B

2 20 0k

0 0

k

•Define a new constant, the Carlson constant:•The equations above simplify to: 0 0

1c

ck

0 0y zE cB 0 0z yE cB

Wave Equations Summarized•Waves look like:•Related by:•Two independent solutions to these equations:

0

0

, , , sin

, , , sin

x y z t kx t

x y z t kx t

E E

B Bck 0 0

0 0

ory z

z y

E cB

E cE

E0

B0

E0

B0

0 0E cB

•Note that E, B, and direction of travel are all mutually perpendicular•The two solutions are called polarizations•We describe polarization by telling which way E-field points

•Note E B is in direction of motion

Understanding Directions for Waves

•The wave can go in any direction you want•The electric field must be perpendicular to the wave direction•The magnetic field is perpendicular to both of them

•Recall: E B is in direction of motion

0 0E cB

The Meaning of c

•Waves traveling at constant speed•Keep track of where they vanish

0

0

, , , sin

, , , sin

x y z t kx t

x y z t kx t

E E

B B

ck

•c is the velocity of these waves

0 0

1c

0kx t x tk

ct

83.00 10 m/sc

82.99792458 10 m/s

•This is the speed of light•Light is electromagnetic waves!•But there are also many other types of EM waves•The constant c is one of the most important fundamental constants of the universe

Wavelength and Wave Number•The quantity k is called the wave number•The wave repeats in time•It also repeats in space

0

0

sin

sin

kx t

kx t

E E

B B

ck 12

f Tf

2k

•EM waves most commonly described in terms of frequency or wavelength

ck

22

f

c f

•Some of these equations must be modified when inside a material

The Electromagnetic Spectrum•Different types of waves are classified by their frequency (or wavelength) c f

Radio WavesMicrowavesInfraredVisibleUltravioletX-raysGamma Rays

Incr

easi

ngf I

ncre

asin

gRed

OrangeYellowGreenBlue

Violet

VermillionSaffron

ChartreuseTurquoise

Indigo

Know these, in order

These tooNot these

•Boundaries are arbitrary and overlap•Visible is 380-740 nm

Energy and the Poynting Vector•Let’s find the energy density in the wave

2102Eu E 2 21

0 02 sin kx t E 2 2 210 02 sinc B kx t 2 20

00 0

1 sin2

B kx t

2

02Bu

B

2

20

0

sin2B

kx t

2

20

0

sinBu kx t

•Now let’s define the Poynting vector: 0

1

S E B

20 0

0

1 sinS E B kx t

cu

0 0E cB

•It is energy density times the speed at which the wave is moving•It points in the direction energy is moving•It represents the flow of energy in a particular direction •Units:

ucS 3

J mm s

2

Wm

Intensity and the Poynting Vector•The time-averaged Poynting vector is called the Intensity

•Power per unit areaS c u

220

0

sincB kx t

20

02cB

S

In Richard Williams’ lab, a laser can (briefly) produce 50 GW of power and be focused onto a region 1 m2 in area. How big are the electric and magnetic fields?

SA

P

10

26

5.0 10 W

10 m

22 25.0 10 W/m

020

2 SB

c

7 22 2

8

2 4 10 T m/A 5.0 10 W/m

3 10 m/s

8 24.2 10 T

0 20,000 TB 0 0E cB 120 6.1 10 V/mE

Momentum and Pressure•Light carries energy – can it carry momentum?

•Yes – but it’s hard to prove•p is the total momentum of a wave and U the total energy

•Suppose we have a wave, moving into a perfect absorber (black body)•As they are absorbed, they transfer momentum

•Intensity:

Upc

US At

•As waves hit the wall they transfer their momentum

cFA

c pSA t

P S c

cP

•Pressure on a perfect absorber:•When a wave bounces off a mirror, the momentum is reversed

•The change in momentum is doubled•The pressure is doubled

2P S c

Cross-Section•To calculate the power falling on an object, all that matters is the light that hits it

•Example, a rectangle parallel to the light feels no pressure•Ask yourself: what area does the light see?

•This is called the cross section

S P = F P= P S c

Sample ProblemA 150 W bulb is burning at 6% efficiency. What is the

force on a mirror square mirror 10 cm on a side 1 m away from the bulb perpendicular to the light hitting it?

•Light is distributed in all directions equally over the sphere of radius 1 m

2

0.06 150 W4 m

S

20.72 W/m

F P2 S

c

2

2

8

2 0.72 W/m0.1 m

3 10 m/s

1 m

114.8 10 NF

Sources of EM Waves•A charge at rest produces no EM waves

•There’s no magnetic field•A charge moving at uniform velocity produces no EM waves

•Obvious if you were moving with the charge•An accelerating charge produces electromagnetic waves

•Consider a current that changes suddenly•Current stops – magnetic field diminishes•Changing B-field produces E-field•Changing E-field produces B-field•You have a wave

+–

Simple Antennas•To produce long wavelength waves, easiest to use an antenna

•AC source plus two metal rods•Some charge accumulates on each rod•This creates an electric field•The charging involves a current

•This creates a magnetic field•It constantly reverses, creating a wave•Works best if each rod is ¼ of a wavelength long•The power in any direction is

– – – – – –

++++++

2

2

sinSr

distan

ce r

Common Sources of EM Waves

Radio Waves AntennasMicrowaves Klystron, MagnetronInfrared Hot objectsVisible Outer electrons in atomsUltraviolet Inner electrons in atomsX-rays Accelerated electronsGamma Rays Nuclear reactions