Short Version : 29. Maxwell’s Equations & EM Waves
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Maxwell Theorem - General Physcis - Lecture Slides, Slides of Physics
This course covers Force, acceleration, gauss law, ideal law, polarization, newton laws, pendulum, reflection and refraction equations, kinetic energy, RC circuits, fusion and many other topics. Key points of this lecture are: Maxwell theorem, Em Waves, Four Laws of Electromagnetism, Ambiguity in Ampere’S Law, Maxwell’S Equations, Maxwell’S Equations in Vacuum, Electromagnetic Waves, Plane Electromagnetic Wave, Gauss’S Laws, Faraday’S Law
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Short Version :
29. Maxwell’s Equations & EM Waves
29.1. The Four Laws of Electromagnetism Law
Mathematical
Statement
PhysicalMeaning
Gauss for
E
q^0
d
^
E
A
How
q
produces
E
;
E
lines begin & end on q’s.
Gauss for
B
Faraday
Ampere (Steady
I^
only)
4 Laws of EM (incomplete)
d^
^
B
A
B
d
d^
d t
^
E
r
0
d^
I
^
B
r
No magnetic monopole; B
lines form loops. Changing
B^
gives emf.
Moving charges give
B
.
Note
E
- B
asymmetry between the Faraday & Ampere laws.
29.3. Maxwell’s Equations
Law
Mathematical
Statement
PhysicalMeaning
Gauss for
E
q^0
d
^
E
A
How
q
produces
E
;
E
lines begin & end on q’s.
Gauss for
B
FaradayAmpere-Maxwell
d^
^
B
A
B
d
d^
d t
^
E
r
No magnetic monopole; B
lines form loops. Changing
B^
gives emf.
Moving charges &changing
E^
give
B
.
0
0
0
E d
d^
I^
d t
^
^
^
^
B
r
Maxwell’s Eqs (1864).Classical electromagnetism.
Maxwell’s Equations in Vacuum
Gauss for
E
d^
^
E
A
Gauss for
B
Faraday Ampere-Maxwell
d^
^
B
A
B
d
d^
d t
^
E
r
0
0
E d
d^
d t
^
^
B
r
Plane Electromagnetic Wave
EM wave in vacuum is transverse:
E
B
k
(direction of propagation)
For uniqueness, see Prob 46
Sinusoidal plane waves going in
x
-direction:
^
^
^
^
,^
y
x t
E
x t
E
j
^
^
^
^
,^
z
x t
B
x t
B
k
ˆ^
k
E
B
^
Right-hand rule
^
^
sin p E
kx
t
^
^
j
^
^
sin p B
kx
t
^
k
Gauss’s Laws
^
^
sin p E
kx
t
^
j
Both
E
&
B
field lines are straightlines,
so their flux over any closed surfaces vanish identically.Hence the Gauss’s laws are satisfied.
Plane wave :
^
^
^
^
,^
y
x t
E
x t
E
j
^
^
^
^
,^
z
x t
B
x t
B
k^
^
^
sin p B
kx
t
^
^
k
Ampere-Maxwell Law
0
0
E
C
d
d^
d t
^
^
B
r
^
^
^
C^
d^
B
x^
h^
B
x^
dx
h
^
^
B
r
I^
For loop at
x
of height
h
& width
dx
^
^
^
,^
,^
B
B
x t
h^
B
x t
d x
h
x
^
^
^
^
B
d x
h
x ^
^
^
E^
E h dx
E d^
E
h dx
d t
t
Ampere-Maxwell Law :
0
0
B
E
x^
t
^
^
^
Ampere-Maxwell law expressed as a differential eq :
0
0
t
^
^
E
B
in vacuum
B
B
z ˆ E E
y
0
0
y
z^
E
B x
t
^
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Conditions on Wave Fields
Faraday’s Law :
E
B
x^
t
^
^
Ampere-Maxwell Law :
0
0
B
E
x^
t
^
^
For
E
E
( x,t
)^
j^
B
B
( x,t
)^
k
For a plane wave
^
^
^
^
,^
sin p
x t
E
kx
t
^
E
j^
^
^
^
^
,^
sin p
x t
B
kx
t
B
k
^
^
^
cos
cos
p^
p
k E
k x
t^
B
k x
t
Faraday’s Law :
p^
p
k E
B
Ampere-Maxwell Law :
^
^
^
0
0
cos
cos
p^
p
k B
k x
t^
E
k x
t
^
^
0
0
p^
p
k B
E
^
x^
y^
z
x^
y^
z
V
V
V
^
^
^
^
^
i^
j^
k
V
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Example 29.2. Laser Light
A laser beam with wavelength 633 nm is propagating through air in the +z direction.Its electric field is parallel to the
x
axis and has magnitude 6.0 kV/m.
Find
(a) the wave frequency,(b) the amplitude of the magnetic field, and(c) the direction of the magnetic field.
c
f
(a)
8
9
m
s m
14
H
z
^
E
B
c
(b) (c)
y axis.
3 8
V
m m
s
5
T
T
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Polarization
Polarization
E
Radiation from antennas are polarized.E.g., radio, TV, ….Light from hot sources are unpolarized.E.g., sun, light bulb, …
Reflection from surfaces polarizes.E.g., light reflecting off car hoods is partially polarized inhorizontal direction.Transmission through crystal / some plastics polarizes.E.g., Polaroid sunglasses, …
Law of Malus :
Only component of
E
// preferred direction
e
is transmitted.
2
2
2 cos
trans
inc
E
E
^
ˆ^
trans
inc
^
E
E
ε^
ε^
cos inc E
ε
^
= angle between
E
inc
&
. 2 cos
trans
inc
S^
S
or
ˆ^
ε^
j^ ˆ ˆ^
ε^
k
Conceptual Example 29.1. Crossed Polarizers
Unpolarized light shines on a pair of polarizers with their transmissionaxes perpendicular, so no light gets through the combination.What happens when a third polarizer is sandwiched in between, with itstransmission axes at 45
^
to the others?
st^
& middle polarizers not
so some light passes through.
Passed light’s polarization not
to axis of last polarizer so
some light passes through.
Making the Connection
How does the intensity of light emerging from this polarizer “sandwich”compare with the intensity of the incident unpolarized light?
2 cos
trans
inc
S^
S
Intensity of light emerging from 1
st^
polarizer :
2
2
1
0 1
cos
inc
S^
S^
d
^
^
^
inc S
^
( polarized along axis of 1
st^
polarizer )
Intensity of light emerging from middle polarizer :
2
2
1
cos 45
S^
S
^
^
1 1 2
S
^
inc S
Intensity of light emerging from ensemble :
2
3
2
cos 45
S^
S
^
^
2 1 2
S
^
inc S
( polarized along axis of middle polarizer.)( polarized along axis of 3
rd^
polarizer.)
29.7. Producing Electromagetic Waves
Any changing
E
or
B
will create EM waves.
Any accelerated charge produces radiation.Radio transmitter: e’s oscillate in antenna driven by
LC
circuit.
X-ray tube: accelerated e’s slammed into target.MW magnetron tube: e’s circle in
B
.
EM wave : f^
=^
f^ of
q
motion
Most efficient:
^
~ dimension of emitter / reciever
Waves emit / receive
axis of dipole.
Sourcereplenishesradiatedenergy
LC oscillatordrives
I^
in
antenna
OutgoingEM waves
29.8. Energy & Momentum in Electromagetic Waves
Consider box of thickness
dx
, & face
A
k
of EM wave.
2 0 1 2
E u
E
Energy densities:
2 0 1 2
u^ B
B
Energy in box:
^
E^
B
dU
u^
u^
A dx
^
2
2
0
0
A
dx
^
^
^
^
E
B
Rate of energy moving through box:
dU
d U
d t
dx
c
2
2
0
0
A
c
^
^
^
^
E
B
Intensity
S
= rate of energy flow per unit area
2
2
0
0
S^
c
^
^
^
^
^
E
B
Plane waves:
E
B
c
0
0
S^
c^
E B c
c
^
^
^
^
^
2
0
0
0 1
c^