1. Electromagnetic Waves. Energy Carried by Electromagnetic Waves.
Today’s agenda:
Electromagnetic Waves.
Energy Carried by Electromagnetic Waves.
Momentum and Radiation Pressure of an Electromagnetic Wave.
rarely in the course of human events have so many starting equations been given in so little time

2. We began this course by studying fields that didn’t vary with time—the electric field due to static charges, and the magnetic field due to a constant current.
In case you didn’t notice—about a half dozen lectures ago things started moving!
We found that changing magnetic field gives rise to an electric field. Also a changing electric field gives rise to a magnetic field.
These time-varying electric and magnetic fields can propagate through space.

3. Electromagnetic Waves
Maxwell’s Equations
These four equations provide a complete description of electromagnetism.

4. Production of Electromagnetic Waves
Apply a sinusoidal voltage to an antenna.
Charged particles in the antenna oscillate sinusoidally.
The accelerated charges produce sinusoidally varying electric and magnetic fields, which extend throughout space.
The fields do not instantaneously permeate all space, but propagate at the speed of light. y
direction of
propagation x z

5. This static image doesn’t show how the wave propagates.y
direction of
propagation x z
This static image doesn’t show how the wave propagates.
Here are animations, available on-line:
(shows electric field only)
Here is a movie.

6. Therefore, electromagnetic waves can propagate in free space.
Electromagnetic waves are transverse waves, but are not mechanical waves (they need no medium to vibrate in).
Therefore, electromagnetic waves can propagate in free space.
At any point, the magnitudes of E and B (of the wave shown) depend only upon x and t, and not on y or z. A collection of such waves is called a plane wave. y
direction of
propagation x z

7. These equations have solutions:
Manipulation of Maxwell’s equations leads to the following plane wave equations for E and B:
Equations on this slide are for waves propagating along x-direction.
These equations have solutions:
Emax and Bmax are the electric and magnetic field amplitudes
where
You can verify this by direct substitution.
Emax and Bmax in these notes are sometimes written by others as E0 and B0.

8. You can also show that
At every instant, the ratio of the magnitude of the electric field to the magnitude of the magnetic field in an electromagnetic wave equals the speed of light.

9. Emax (amplitude) E(x,t) direction of propagationy
direction of
propagation x z
Emax (amplitude)
E(x,t)
Nit-picking detail: Ex is the x-component of the electric field vector. It has both a magnitude and a direction. E is the magnitude of the electric field vector, and because for this wave the electric field has only an x-component, E =  Ex . I may get careless in class and refer to Ex as the magnitude of E, when I should really include the absolute values:  Ex .