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29. Maxwell’s Equations
Topics

Laws of Electric & Magnetic Fields

James Clerk Maxwell

Maxwell’s Equations
2
Laws of Electric & Magnetic
Fields
3
James Clerk Maxwell
1831 – 1879
In 1865, Maxwell published a
paper entitled:
A Dynamical Theory of the
Electromagnetic Field,
Philosophical Transactions of
the Royal Society of London
155, 459-512 (1865).
This is one of the greatest
scientific papers ever written.
4
Maxwell’s Equations

E  dA 
Qinside
Closed Surface

0
B  dA  0
Closed Surface
d m
E  dr  

dt
Closed Loop
5
de
B  dr  0 I  0 0

dt
Closed Loop
Displacement Current
Maxwell realized that Ampere’s law is not
valid when the current is discontinuous as
is true of the current through a parallel
plate capacitor:

B  dr  0 I Encircled
Closed Loop
wikimedia.org
6
Displacement Current
He concluded that when the charge within an enclosed
surface is changing it is necessary to add to Ampere’s
law another current called the
displacement current: ID
dQinside
de
ID 
 0
dt
dt

B  dr  0 ( I  I D )
Closed Loop
7
wikimedia.org
The
nd
2
Unification of Forces
The 2nd Unification of Forces
0 is the electric
 0  8.854 10 C /(N  m )
12
0 is the magnetic
constant
2
2
constant
0  4 107 N/A 2
0 0  4 10 [ N/A ]
7
2
12
 8.854 10 [C /(N  m )]
17
2
2
 1.113 10 [s /m ]
9
2
2
The 2nd Unification of Forces
From
0 0  1.113 1017 [s 2 /m 2 ]
we can write
1
0 0
 2.998 10 m/s
8
which is the speed of light in vacuum!
10
Light
“We can scarcely avoid
the conclusion that light
consists in the transverse
undulations of the same
medium which is the cause
of electric and magnetic
phenomena.” (1866)
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2nd Unification
3rd Unification
4th Unification?
5th Unification?
1st Unification
12
Summary

13
Maxwell’s Equations
 Gauss’s Law for E
 Gauss’s Law for B
 Faraday’s Law
 Ampere’s Generalized Law
Electromagnetic Waves
Topics

Maxwell’s Wave Equations

Waves – Recap

Electromagnetic Waves

Electromagnetic Radiation
15
Maxwell’s Wave Equations
 E 1  E Wave equation
for E

2
2
2
x
c t
2
2
1  B
Wave equation  B
 2 2
for B
2
x
c t
2
2
These equations describe electric and
magnetic waves traveling in the x direction
16
Maxwell’s Wave Equations
Maxwell showed that the different components of
the electric and magnetic fields are related:
Ez By

x
t
Relationship between
Bz and Ey
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Relationship between
Ez and By
E y
Bz

t
x
Waves – Recap
Wave number
Stationary wave
y ( x)  A sin(kx)
k
2

Wave traveling in x direction
y ( x)  A sin k ( x  vt )
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2
  kv 
T
Electromagnetic Waves
Consider an electric wave, traveling in the
positive x direction, but oscillating in the y direction:
E y ( x, t )  Ep sin(kx  t )
We can find Bz from
Ep
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E y
Bz

t
x
Electromagnetic Waves
This leads to the result
Bz ( x, t )  Bp sin(kx  t )
where
z
Bp  (k /  ) Ep
that is,
Bp
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Ep  cBp
Electromagnetic Waves
Electromagnetic waves always travel in the direction of
the
ˆ
E  Ep sin(kx  t ) j
Poynting vector:
B  Bp sin(kx  t ) kˆ
S
y
EB
0
Units: W/m2
x
z
21
Electromagnetic Waves
But the direction of the electric and magnetic fields
themselves, that is, their polarization, can change
Linear polarization
y
x
z 22
Polarizers
Only a component
Epcos of the
electric field
along the
polarization
axis can get through
1
Law of Malus
S  S0 cos 2 
23
2
3
90o
Electromagnetic Radiation
The
Electromagnetic
Spectrum
25
Spectral Response
human
bee
butterfly
http://landsat.gsfc.nasa.gov/education/compositor
26
Himalyan balsam
Electromagnetic Radiation
An electromagnetic wave carries
energy and momentum.
The average power per unit area is
called the intensity of the wave
The momentum per unit time (that is, force)
per unit area is called the radiation pressure
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Electromagnetic Radiation
The radiation pressure, Prad, is given by
S
Prad 
c
where the average intensity is given by
1 Ep Bp
S

0 2 0
EB
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which can be written in
terms of energy density:
S  cuE  cuB
The Pressure of Sunshine
Solar Luminosity
L
= 3.8 x 1026 W
Astronomical Unit
r
= 1.5 x 1011 m
Intensity
S
= L / 4  r2
Pressure
P
=S/c
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The Pressure of Sunshine
Intensity S
= L / 4  r2
= 1370 W/m2
Pressure
=S/c
= 4.6 N/m2
30
P
Interstellar Travel
31
Credit: Michael Carroll, The Planetary Society
Summary


32
Maxwell’s Equations
 2nd Unification of forces
 Electromagnetic waves
 Universal speed c = 3 x 108 m/s
Electromagnetic Waves
 Gamma rays to radio waves
 Carry energy and momentum
 Exert pressure