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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
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.
Electromagnetic Waves
Maxwell’s Equations
q enclosed
 E  dA  o
d B
 E  ds   dt
 B  dA  0
dΦ E
 B  ds=μ 0 Iencl +μ 0ε 0 dt
These four equations provide a complete description of
electromagnetism.

E 
0
 B  0
dB
×E=dt
1 dE
  B= 2
+μ 0 J
c dt
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
x
z
direction of
propagation
y
x
direction of
propagation
z
This static image doesn’t show how the wave propagates.
Here are animations, available on-line:
http://phet.colorado.edu/en/simulation/radio-waves
(shows electric field only)
http://www.phy.ntnu.edu.tw/ntnujava/index.php?topic=35
Here is a movie.
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
x
z
direction of
propagation
Manipulation of Maxwell’s equations leads to the following
Equations on this slide are for waves
plane wave equations for E and B:
propagating along x-direction.
 2E y
x
2
= 0 0
 2E y (x, t)
 2B z
 2B z (x,t)
= 0 0
2
x
t 2
t 2
These equations have solutions:
Ey =Emax sin kx - t 
Bz =Bmax sin kx - t 
where
2
k= ,

 = 2f ,
and
Emax and Bmax are the
electric and magnetic
field amplitudes

f = = c.
k
You can verify this by direct substitution.
Emax and Bmax in these notes are sometimes written by others as E0 and B0.
You can also show that
E y
B z
=x
t
 Emax sin kx - t 
x
=-
 Bmax sin kx - t 
t
Emax k cos kx - t  =Bmax  cos kx - t 
Emax E 
1
= = =c=
.
Bmax B k
0 0
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.
y
direction of
propagation
x
z
Emax (amplitude)
E(x,t)