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Chapter 24
Classical Theory of
Electromagnetic Radiation
Maxwell’s Equations
Four equations (integral form) :
Gauss’s law

 E  nˆdA 
q
inside
0
Gauss’s law for magnetism
Faraday’s law
Ampere-Maxwell law
+ Lorentz force


 
d 
 E  dl   dt  B  nˆdA
 
d elec 

 B  dl  0  I inside_ path   0 dt 


 
F  qE  qv  B
Fields Without Charges
Time varying magnetic field makes electric field
Time varying electric field makes magnetic field
Do we need any charges around to sustain the fields?
Is it possible to create such a time varying field configuration
which is consistent with Maxwell’s equation?
Solution plan: • Propose particular configuration
• Check if it is consistent with Maxwell’s eqs
• Show the way to produce such field
• Identify the effects such field will have on matter
• Analyze phenomena involving such fields
A Simple Configuration of Traveling Fields
Key idea: Fields travel in space at certain speed
Disturbance moving in space – a wave?
1. Simplest case: a pulse (moving slab)
Note: strictly speaking fields don’t
move, they just change in time
A Pulse and Gauss’s Laws

 E  nˆdA 
q
inside
0

 E  nˆdA  0
Pulse is consistent with Gauss’s law

 B  nˆA  0
Pulse is consistent with Gauss’s law
for magnetism
A Pulse and Faraday’s Law
emf  
d mag
dt
Since pulse is ‘moving’, B depends
on time and thus causes E
 mag  Bhv t
mag d mag

 Bhv
t
dt
emf
 
emf   E  dl  Eh
Is direction right?
Area does
not move
E=Bv
A Pulse and Ampere-Maxwell Law
=0
 
d elec 

 B  dl  0  I inside_ path   0 dt 
elec  Ehvt
 elec d elec

 Ehv
t
dt
 
 B  dl  Bh
Bh  0 0 Evh
B  0 0vE
A Pulse: Speed of Propagation
B  0 0vE
E=Bv
B  0 0vBv
1  0 0v 2
v
1
0 0
 3  108 m/s
E=cB
Based on Maxwell’s equations, pulse must propagate at speed of light
Clicker
In a time t, what is mag?
A) 0; B) Bvt; C) Bhvt; D) Bxh; E) B(x+vt)h
Clicker
d mag
emf 
 Bvh
dt
r r
—
 Egdl  Eh
What is E?
A) Bvh; B) Bv; C) Bvh/(2h+2x); D) B; E) Bvh/x
Exercise
If the magnetic field in a particular pulse has a magnitude of
1x10-5 tesla (comparable to the Earth’s magnetic field), what is
the magnitude of the associated electric field?
E  cB
E  3x108 m / s  1x105 T  3000V / m
Force on charge q moving with velocity v perpendicular to B:
Fmag/Fele = qvB/qE = vB/cB=v/c
Direction of Propagation
Direction of speed is given
by vector product
 
EB
Electromagnetic Radiation
Electromagnetic Spectrum
Maxwell’s Theory of Electromagnetism
• Light is electromagnetic wave!
• Challenge: Design an electric
device which emits and detects
electromagnetic waves
(1831-1879)
Accelerated Charges
Electromagnetic pulse can propagate in space
How can we initiate such a pulse?
Short pulse of transverse
electric field
Accelerated Charges
1. Transverse pulse
propagates at speed of
light
2. Since E(t) there must
be B
3. Direction
 of v is given
by: E  B
E
v
B
Accelerated Charges: 3D
Magnitude of the Transverse Electric Field
We can qualitatively predict the direction.
What is the magnitude?
Magnitude can be derived from
Gauss’s law
Vectors a, r and E always in one plane
Field ~ -qa

Eradiative 

1  qa
40 c 2 r
1. The direction of the field is opposite to qa
2. The electric field falls off at a rate 1/r
Exercise
a
An electron is briefly accelerated in the
direction shown. Draw the electric and
magnetic vectors of radiative field.
E
B
1. The direction of the field is opposite to qa
 
2. The direction of propagation is given by E  B
Exercise
An electric field of 106 N/C acts on an electron
for a short time. What is the magnitude of
electric field observed 2 cm away?
1. Acceleration a=F/m=qE/m=1.78.1017 m/s2
E=106 N/C
B
2 cm
a
2. The direction of the field is opposite to qa

3. The magnitude: 
1  qa
.10-7 N/C
E=1.44
Eradiative 
40 c 2 r
 
4. The direction of propagation is given by E  B
Erad
What is the magnitude of the Coulomb field at the same location?
q
6
E 

3
.
6

10
N/C
2
40 r
1
Question
A proton is briefly accelerated as shown below. What
is the direction of the radiative electric field that will
be detected at location A?
B
A
A
+
D
C