Download Chapter 3 Electromagnetic Theory, Photons, and Light

Lecture 6
Chapter 3
Electromagnetic Theory, Photons,
and Light
 Photons
 Radiation
 Emission of light by atoms
Example problem
A laser pointer emits light at 630 nm in xy plane at =450 to axis x
(counter clock-wise). The light is polarized along axis z , beam
cross-section is A=1 mm2 and its power is P=1 mW.
1. Write an equation of E and B components of this EM wave for the
region of the beam.
z E
 
 
E  E0 cos k  r  t
y
B
Find :   2  2c 
Find k: k  2   iˆ cos   ˆj sin 
Find E0:
P
c 0 2
x


I
E0
Irradiance:
A
2

E0  2 P Ac 0
E0  2 P Ac 0 k̂




Electric field: E 



 2c 
2P
 2 ˆ
ˆ
k̂ cos 
i cos   j sin   r 
t
 
Ac 0

Example problem (continued)

E


 2c 
2P
 2 ˆ
ˆ
k̂ cos 
i cos   j sin   r 
t
 
Ac 0

 
 
Magnetic field: B  B0 cos k  r  t
It is in phase with E.
Need only find its amplitude and direction.

B0  E0 / c 
 1
B0 
c
 1
B
c
1
c

y
B
x
2P
Ac 0

2P ˆ
i sin   ˆj cos 
Ac 0

z




 2c 
2P ˆ
 2 ˆ
ˆ
ˆ
i sin   j cos  cos 
i cos   j sin   r 
t
Ac 0
 

Example problem (continued)
2. This laser beam is reflected backwards by a mirror. What is
the average force on the mirror due to the radiation pressure?
Find average pressure
Pt  T
Find force: F  Pt  T
P
10 3 W
12
A2 2

6
.
6

10
N
8
c
3  10 m/s
I
P
2 2
c
Ac
3. How much energy is contained in EM field of 1 m long beam?
Power is amount of energy per unit time. During one second,
light travels c meters:
P
103 J/s
12
Energy (1m )  L 
1
m

3
.
3

10
J
8
c
3  10 m/s
Alternatively can find u using E0 and multiply by volume
Classical EM waves versus photons
The energy of a single light photon is E=h
c
The Planck’s constant h = 6.626×10-34 Js
E1  h  h  4  1019 J
Visible light wavelength is  ~ 0.5 m

Example: laser pointer output power is ~ 1 mW
number of photons emitted every second:
P
103 J/s
15


2
.
5

10
photons/s
19
E1 4  10 J/photon
Conclusion: in many every day situations the quantum nature of
light is not pronounced and light could be treated as a classical
EM wave
Photons
1900: to explain black body radiation spectrum Max Planck
suggested that light is emitted in small indivisible quanta of energy:
E=h
(h=6.626×10-34 J.s)
1905: to explain photoelectric effect Einstein stated that EM field
itself is quantized
Photons cannot be observed directly, one can only see them through
interaction with matter - absorption. Photon is destroyed in the
process.
Photons carry energy and momentum (atoms recoil when emit photons)

h

or
,
where

p
p  k

2
h
propagation vector
Experimental confirmation:
Compton effect
Photon
Mean photon flux:  
no. photons per unit area per unit time
AI
P

h 0 h 0
Energy per unit time
crossing some area A
optical power
Energy of a single photon at
mean frequency 0 in quasimonochromatic beam
Number of photons emitted every second from a ~1 mW laser
pointer is ~1015 photons
Photons strike screen every 1 fs on average.
Exact position and time of arrival for each photon cannot be predicted
with absolute certainly - we can only predict the probabilities.
Photon
low
Light exposure
medium
high
photographic film
At any location on a screen, the classical irradiance is proportional
to the probability of detecting a photon at that location
Quantum uncertainty.
Example: throw a single coin, it will fall either heads or tails up, unpredictable
but with 1015 coins - can predict result with high precision
Photon statistics
Maxwell-Boltzmann statistics: for distinguishable particles
In quantum physics for indistinguishable particles:
* Bose-Einstein statistics for bosons (particles with integer spin)
* Fermi-Dirac statistics for fermions (particles with integer+half spins)
Photons are bosons - many photons can simultaneously be in
exactly the same state, i.e. have the same energy
When a large number of photons occupy the same state (i.e. have the
same energy, polarization and direction), the inherent granularity of
the light beam vanishes and the EM field appears as the continuous
medium of an electromagnetic wave - monochromatic plane wave.
Different monochromatic plane waves represent different photon states
Photon counter
It is possible to detect single photons
Example: photomultiplier tube (PMT)
Photon kicks an electron out of
cathode
The electron is accelerated by an
E-field toward a dynode
The accelerated electron strikes
the dynode and kicks out more
electrons
Many dynodes are used to get
burst of ~105 electrons per single
photoelectron
The burst of electron current can
be detected electronically
Photon statistics
low power
light beam
PMT
Photons arrive at random.
Poisson distribution of photons
arriving at detector during time T
Radiation: accelerated charges
Field of a moving charge
Electromagnetic pulse can propagate in space
How can we initiate such pulse?
Short pulse of transverse
electric field
Radiation: 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
Electric dipole radiation
Oscillating charges in dipole create sinusoidal E
field and generate EM radiation
Electric dipole radiation
Dipole moment:
p  qd
d  d 0 cost 
p  p0 cost 
Electric field of oscillating dipole:
p0k 2 sin  coskr  t 
E
4 0
r
Irradiance:
2

p0   4
I   
2 3
sin 2 
32 c  0 r 2
* EM wave is polarized along dipole
* I ~ 4 - higher frequency, stronger radiation
* No radiation emitted in direction of dipole
Dipole antenna
Example: connect AC
generator to ‘dipole’ antenna/
Charges will run up and
down - dipole moment will
be oscillating and radiation
will be emitted