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.

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

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:

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:

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

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

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

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

The Electromagnetic Spectrum Different types of waves are classified by their frequency (or wavelength) Boundaries are arbitrary and overlap Visible is 380-740 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

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:

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?

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

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

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

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 – – – – – – ++++++  distance r

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