MAXWELL’S EQUATIONS AND
ELECTROMAGNETIC
RADIATION
Purdue University – Physics 241 – Lecture 21
Brendan Sullivan
Overview

Maxwell’s Displacement Current

Maxwell’s Equations

Wave Equation for Electromagnetic Waves

Electromagnetic Radiation
 EM
Spectrum
 Momentum and Pressure from EM Waves
Maxwell’s displacement current
corrects Ampere’s Law

-Q

Q
E
E
0


Ampere’s Law:
 
B d

I
0 enc
Consider a charging capacitor
Apply Ampere’s Law S1 and
S2; we should get the same
answer
 
Current I passes through
S1: B d 
0I
 
S2:
B d  0 Current stops at plate
Maxwell generalized
Ampere’s Law to account for
this with displacement current
A time varying electric flux creates a
displacement current and thus a B field



Maxwell: A changing electric
field induces a magnetic field
Faraday: A changing
magnetic field induces an
electric field
We can update Ampere’s
Law to reflect this induced B
field!
 
B d
I
0 enc
0 0
Id
0
d E
dt
Q Q
E
E 0
d E
dt
Maxwell’s equations describe all of
E&M (for PHYS241 at least)
S
 
E n dA
qenc

B ndA 0
S


0
C
C
 
E d

B d
d B
dt
0
( I enc I d )
Everything you’ve learned this semester (and much
more) boils down to these equations
One important application, which we’ll explore for
the rest of the semester, is electromagnetic waves
 We won’t derive the wave equation from these in
class, but it can be done (I’ll add a supplement)
The wave equation describes all waves,
including EM waves

In deriving EM waves, you get a wave equation for
E and a separate for B

E
x2
2

1
c2
2

E
t2

B
x2
2
1
c2
2

B
t2
We can learn a lot about EM waves by thinking
about these equations physically
 They
are “equations of motion” – they relate how
motion must happen with respect to time
 Move
E
at velocity c = 3x108 m/s
and B should behave very similarly because they are
governed by the same equation
Electric fields, Magnetic fields, and
propagation are all perpendicular
1) E and B are perpendicular at all points
3) EM waves propagate (travel) in the ExB direction
2) E and B are in-phase (maxima and minima at same time)
http://www.youtube.com/watch?v=4CtnUETLIFs
Note: This is a linearly polarized wave
Wave Terminology
For transverse waves, the velocity is
related to frequency and wavelength
v
c
f
For EM waves, one can show that the
amplitudes are related by


|E| c|B|
Em
Note that ALL
electromagnetic waves
(e.g. x-rays, visible, UV,
radio, etc.) travel at the
speed of light, c!!!
Quiz Question 1

An EM wave is propagating in the +y direction. The
electric field is parallel to the +z direction. In which
direction must the magnetic field point?
a) +x
b) –x
c) –y
d) +z
e) -z
The Electromagnetic Spectrum
f
c
All of the forms of radiation
we’re about to discuss are
nothing more than
electromagnetic waves
We’ll show later that higher
frequency waves have more
energy; typically anything
UV or higher is in the
DANGER ZONE!!!!
One way to produce EM waves is
through dipole radiation


Electric dipole radiation: Figure 30-7 in book. By
varying the electric field, we can create a
sinusoidal, propagating E field
Magnetic fields:
We will discuss other methods to get EM
waves later in the semester, and you
probably already have in other courses
(e.g. chemistry)
Quiz Question 2
Which of the following could describe an
electromagnetic wave in vacuum?
a) λ = 100 meters, f = 1x106 Hz
b) λ = 105 meters, f = 3x104 Hz
c) λ =10-6 meters, f = 3x1014 Hz
d) λ = 1 meter,
f = 3/2 x 108 Hz
Recall the speed of light, c = 3x108m/s
Electromagnetic waves have some
energy associated with them

We know the energy densities for electric and
magnetic fields
uE

1
2
 2
0 |E|
uB
 2
|B|
1
2
0
Electromagnetic waves should have the energy of
the electric and magnetic field combined


 Also, the magnitudes of E and B are related by | E | c | B |
uEM
u E uM
1
2
 2
0 |E|
1  2 1
| B|
2 0
2
 2
0 |E|

1 E 2
| |
2 0 c
 2
|
0 |E|

B
0
|2
Intensity is the average power incident
on a unit area by an EM wave

Intensity: average rate of energy flow (power) per
unit area
 Energy
is moving with the fields at c, so I uav c
 To get the average, energy consider rms E and B fields
I
u av c
1 Em axBm ax 
|S|
2
0
Erms Brms
0

S is known as the Poynting vector; it describes wave
intensity (and points which way the wave is going)

S
1  
E B
0
Electromagnetic waves carry
momentum!

Light transfers energy; a change in energy comes
from work; work comes from forces; forces change
momentum
U
 In

short: EM waves actually push on things.
c
If we consider this force applied over an area, light
exerts a pressure when it hits something. We call
this radiation pressure
Pr

p
I
c
E0 B0
2 0c
E02
2 0c 2
B02
2 0
Here, we are talking about light incident on
perfectly absorbing surfaces; the radiation pressure
is doubled for perfectly reflecting surfaces
Really, really sweet application: Solar
Sails

Idea: use the sun’s radiation pressure as a driving
force to propel a ship through space without fuel
Example: Solar Sail
The sun radiates power at a rate of 3.8x1026J/s (P =
3.8x1026W). If we have a solar sail 1km2 near the
Earth’s surface, determine its acceleration. The shit
weighs 10kg and RE = 1.52x1011M.