4. 4
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.
6. 6
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:
Encircled
Closed Lo
0
op
B dr Iµ× =∫
r r
wikimedia.org
7. 7
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
inside
0D
edQ
I
t dtd
dϕ
ε= =
wikimedia.orgD
Closed Loop
0 ( )B dr I Iµ× = +∫
r r
9. 9
The 2nd
Unification of Forces
µ0 is the magnetic
constant
7 2
0 4 10 N/Aµ π −
= ×
ε0 is the electric
constant
12 2 2
0 8.854 10 C /(N m )ε −
= × ×
7
0 0
12
1
2
2 2
27 2
4 10 [ ]
8.854 10 [ ]
1.1
N/A
13
C /(N m )
s10 [ ]/m
µ ε π −
−
−
= ×
× ×
= ×
×
10. 10
The 2nd
Unification of Forces
8
0 0
1
2.998 10 m/s
µ ε
= ×
From
7
0
2
0
21
1.113 1 s /m0 [ ]µ ε −
= ×
we can write
which is the speed of light in vacuum!
11. 11
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)
16. 16
Maxwell’s Wave Equations
2 2
2 2 2
1E E
x c t
∂ ∂
=
∂ ∂
r r
Wave equation
for E
2 2
2 2 2
1B B
x c t
∂ ∂
=
∂ ∂
r r
Wave equation
for B
These equations describe electric and
magnetic waves traveling in the x direction
17. 17
Maxwell’s Wave Equations
yz
BE
x t
∂∂
=
∂ ∂
Relationship between
Ez and By
yz
EB
t x
∂∂
= −
∂ ∂
Relationship between
Bz and Ey
Maxwell showed that the different components of
the electric and magnetic fields are related:
18. 18
Waves – Recap
( ) sin( )Ay x kx=
( ) sin ( )y x k x tA v= −
Stationary wave
Wave traveling in x direction
Wave number
2
k
λ
π
=
2
kv
T
π
ω = =
19. 19
Electromagnetic Waves
p( , ) sin( )yE x t kE x tω= −
Consider an electric wave, traveling in the
positive x direction, but oscillating in the y direction:
We can find Bz from
yz
EB
t x
∂∂
= −
∂ ∂
Ep
20. 20
Electromagnetic Waves
p( , ) sin( )zB x t kB x tω= −
This leads to the result
where
p p( / )kB Eω=
p pE cB=
Bp
z
that is,
21. 21
y
x
z
Electromagnetic Waves
p
p
ˆsin( )
ˆsin( )
E kx t j
B kx
E
B t k
ω
ω
= −
= −
r
r
Electromagnetic waves always travel in the direction of
the
Poynting vector:
0
E B
S
µ
×
=
r r
r
Units: W/m2
22. 22
Electromagnetic Waves
But the direction of the electric and magnetic fields
themselves, that is, their polarization, can change
y
x
z
Linear polarization
23. 23
Polarizers
1
2
3 90o
2
0 cosS S θ=
Law of Malus
Only a component
Epcosθ of the
electric field
along the
polarization
axis can get through
27. 27
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
28. 28
Electromagnetic Radiation
The radiation pressure, Prad, is given by
rad
S
P
c
=
where the average intensity is given by
p p
0 0
1
2
E BEB
S
µ µ
= =
which can be written in
terms of energy density:
E BS cu cu= =
29. 29
The Pressure of Sunshine
Solar Luminosity L = 3.8 x 1026
W
Astronomical Unit r = 1.5 x 1011
m
Intensity S = L / 4π r2
Pressure P = S / c
30. 30
The Pressure of Sunshine
Intensity S = L / 4π r2
= 1370 W/m2
Pressure P = S / c
= 4.6 µN/m2
32. 32
Summary
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
