1. Electromagnetic Waves
Wave Solutions
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:
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.

2. Electromagnetic waves are transverse
Does It Satisfy Gauss’s Laws?
E depends only on x, so there’s only one term in this derivative:
This implies E0x = 0
Same argument applies for magnetic fields
Note that the electric and magnetic fields are perpendicular to the direction the wave is traveling
Electromagnetic waves are transverse

3. Does it Satisfy Faraday’s Law?
All terms in the first equation vanish (Bx = 0)
The others are non-trivial:
Similarly, from the third equation:

4. Does it Satisfy Ampere’s Laws
Very similar calculations to previous slide
Multiply each of these equations
Define a new constant, the Carlson constant:
The equations above simplify to:

5. Wave Equations Summarized
Waves look like:
Related by:
Two independent solutions to these equations: E0 B0
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 E0 B0
Note E  B is in direction of motion

6. 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

7. The Meaning of c Waves traveling at constant speed
Keep track of where they vanish
c is the velocity of these waves
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

8. Wavelength and Wave Number
The quantity k is called the wave number
The wave repeats in time
It also repeats in space 
EM waves most commonly described in terms of frequency or wavelength
Some of these equations must be modified when inside a material

9. The Electromagnetic Spectrum
Different types of waves are classified by their frequency (or wavelength)
Boundaries are arbitrary and overlap
Visible is nm
Radio Waves
Microwaves
Infrared
Visible
Ultraviolet
X-rays
Gamma Rays
f Increasing
 Increasing
Red
Orange
Yellow
Green
Blue
Violet
Vermillion
Saffron
Chartreuse
Turquoise
Indigo
Not these
Know these, in order
These too

10. Energy and the Poynting Vector
Let’s find the energy density in the wave
Now let’s define the Poynting vector:
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:

11. Intensity and the Poynting Vector
The time-averaged Poynting vector is called the Intensity
Power per unit area
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?

12. 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:
As waves hit the wall they transfer their momentum
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

13. 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

14. Sample Problem
A 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?
1 m
Light is distributed in all directions equally over the sphere of radius 1 m

15. 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 + –

16. 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
– – – – – –
++++++ 
distance r

17. Common Sources of EM Waves
Radio Waves Antennas
Microwaves Klystron, Magnetron
Infrared Hot objects
Visible Outer electrons in atoms
Ultraviolet Inner electrons in atoms
X-rays Accelerated electrons
Gamma Rays Nuclear reactions