Download Electromagnetic Theory, Photons and Light • Introduction – Maxwell

Electromagnetic Theory, Photons and Light
• Introduction
– Maxwell etc. showed that light is most certainly electromagnetic in
nature. Classical electrodynamics leads to the picture of a continuous
transfer of energy by electromagnetic waves.
– Quantum electrodynamics describes electromagnetic interactions and
the transport of energy in terms of massless elementary particles called
photons.
– If the wavelength of light is small comapred to the size of the apparatus
- geometrical optics. Otherwise its Physical optics.
– Concepts of waves and particles must be amalgamated.
– Basic laws of Electromagnetic Theory
∗ If a point charge q experiences a force F E , the electric field, E is
defined by F E = q.E.
∗ Observe that a moving charge experiences another force, F M proportional to its velocity v.
∗ Led to the existence of another field, the magnetic field B, such
that F M = q.v × B. Then the Lorentz force is
F = qE + qv × B.
∗ Electric fields are generated by both electric charges and timevarying magnetic fields.
∗ Magnetic fields are generated by electric fields and time-varying
electric fields.
∗ A changing magnetic fields generates a current,
dΦM
,
dt
where ΦM is the instantaneous magnetc flux associated with a
circuit. This is Faraday’s law of induction.
emf = −
ΦM =
Z Z
A
B.dS.
∗ Since field is the potential gradient,
emf =
or
Z
C
d
E.dl = −
dt
C
Z
Z
C
E.dl = −
1
E.dl,
Z Z
A
Z Z
A
B.dS,
∂B
.dS.
∂t
∗ A time varying magnetic field will have an electric field associated
with it.
∗ Gauss’ law - electric:
Z Z Z
Z Z
1
E.dS =
ρdV,
0
A
V
where 0 = 8.8542 × 10−12 C 2 /N.m2 is the electric permittivity of
free space.
∗ Its medium dependent, so in general,
= KE 0 ,
where KE is the dielectric constant.
∗ Gauuss’ law - magnetic:
∗ Flux of magnetic field ΦM through a closed surface is zero,
ΦM =
∗ Amperes law:
Z
C
where
Z Z
A
B.dl = µ
Z Z
B.dS = 0.
A
(J + ∂E
.dS,
∂t
J D = ∂Eover∂t,
is the displacement current.
∗ Maxwell’s Equations
∗ Previously knew Maxwell’s equations as
I
q
E.dA = ,
0
I
B.dA = 0,
dΦB
,
dt
I
dΦE
.
B.ds = µ0 I + 0 µ0
dt
∗ Consider free space, where = 0 , µ = µ0 and ρ = 0, J = 0. Then
I
E.ds = −
∂B
.dS,
A ∂t
C
Z
Z Z
∂E
.dS,
B.dl = µ0 0
C
A ∂t
Z Z
Z
E.dl = −
A
Z Z
A
2
Z Z
B.dS = 0,
E.dS = 0.
∗ E and B appear in the equations with a lot of symmetry.
∗ However E affects B, B affects E.
∗ In general, E and B can be written as
E = (Ex , Ey , Ez ),
B = (Bx , By , Bz ).
∗ In differential form,
∂E z ∂E y
∂B
−
= − x,
∂y
∂z
∂t
∂B
∂E x ∂E z
−
= − y,
∂z
∂x
∂t
∂E y ∂E x
∂B
−
= − z,
∂x
∂y
∂t
∂B z ∂B y
∂E x
−
= µ0 0
,
∂y
∂z
∂t
∂E y
∂B x ∂B z
−
= µ0 0
,
∂z
∂x
∂t
∂B y ∂B x
∂E z
−
= µ0 0
,
∂x
∂y
∂t
∂B x ∂B y ∂B z
+
+
= 0,
∂x
∂y
∂z
∂E x ∂E y ∂E z
+
+
= 0.
∂x
∂y
∂z
∗ Transformed Maxwell’s equations in terms of integrals over finite
regions to derivatives at points in space.
– Electromagnetic Waves
∗ Note the symmetry and interdependence of E and B in Maswell’s
equations.
∗ A time varying E field generates a B field which is everywhere
perpendicular to the direction in which E changes.
∗ A time varying underlineB field generates an E field which is
everywhere perpendicular to the direction in which B changes.
∗ An electromagnetic wave consists of oscillating E and B fields
which are oscillating perpendicular to each other and to the direction of propogation.
∗ Consider a charge that is caused to accelerate from rest.
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∗ When the charge is at rest, it has associated with it a constant
radial E field extending in all directions to infinity.
∗ At the instant the charge begins to move, the E field is altered in
the vicinity of the charge and this alteration propogates out into
spaceat some finite speed.
∗ The time varying electric field induces a magnetic fiela because of
Maxwell’s equations.
∗ If the charge’s velocity is constant, the rate of change of the E is
steady, and the resulting B field is constant.
∗ But if the charge is accelerating, then ∂E
is not constant and so
∂t
the induced B is time-dependent.
∗ The time varying B field generates an E field and so the process
continues.
∗ As one field changes, it generates another and the pulse moves out
from one point to the next through space.
∗ The E and B fields are two aspects of a single underlying phenomenon - the electromagnetic field, whose source is moving charge.
∗ The electric and magnetic fields generate each other in an endless
cycle.
∗ Electromagnetic waves are transverse.
∗ Maxwell’s equations for free space can be manipulated to give
∇2 E = 0 µ0
∂2E
,
∂t2
∂2B
∇ B = 0 µ0 2 .
∂t
∗ In Catreisan coordinates and writing as scalar equations,
2
∂ 2 Ex
∂ 2 Ex ∂ 2 Ex ∂ 2 Ex
+
+
=
µ
.
0 0
∂x2
∂y 2
∂z 2
partialt2
∗
∗
∗
∗
and similar expressions for Ey , Ez , Bx , By , Bz .
Such expressions have been studied before - the wave equation √
if v = 1/ 0 µ0 .
This is what Maxwell did. Nowdays, µ0 = 4π × 10−7 m.kg/C 2 in
SI units, and 0 = 8.85 × 10−12 s62.C 2 /m3 .kg.
This leads to v = 3 × 108 m/s.
This agreed with previous measurements of the speed of light, so
Maxwell concluded that ”This velocity is so nearly that of light
[from work by Fizeau in 1849], that it seems we have strong reason
to conclude that light itself (including radiant heat, and other
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radiations if any) is an electromagnetic disturbance in the form of
waves propogated through the electromagnetic field according to
electromagnetic laws”.
∗ Light is an electromagnetic wave and travels through vacuum with
a speed that is denoted by c and is equal to 2.99792458 × 108m/s.
∗ For this course, unless specified otherwise, use c = 3 × 108 m/s.
– Transverse Waves
∗
∗
∗
∗
∗
∗
∗
∗
∗
Consider a plane wave propogating in the x direction.
Then E = E(x, t).
But from Maxwell’s equations, ∂Ex /∂x = 0.
Then Ex = constant, but then it cannot be a travelling wave.
Then Ex = 0: the electromagnetic wave has no electric field component in the direction of propogation. That is the E field associated with the plane wave is transverse.
In free space, the plane electromagnetic wave is transverse.
Specifying the moment-by-moment direction of E means knowing
the polarization of the light.
A plane or linearly polarized wave is such that the direction of the
vibrating E vector is fixed.
Can make it so that this direction is parallel to the y - axis, where
E = Ey (x, t)j.
∗ Also know that
∂Bz
∂Ey
=−
∂x
∂t
∗ Therefore Bx , By are constant.
∗ Thus in free space, the plane electromagnetic wave is transverse.
∗ Can write
Ey (x, t) = E0y cos[ω(t − x/c) + ],
the speed of propogation being c.
∗ Then
Z
∂Ey
Bz = −
dt.
∂x
∗ Then we have,
1
Bz (x, t) = E0y cos[ω(t − x/c) + ].
c
∗ It is evident that,
Ey = cBz .
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∗ It follows from this that E and B are in phase at all points in
space.
– Energy and Momentum
∗ Electromagnetic waves transport energy and momentum.
∗ Consider u, the energy density or radiant energy per unit volume.
∗ Energy density of an E field alone is
uE =
0 2
E .
2
∗ Similarly the energy density of a B field alone is
uB =
1 2
B .
2µ0
∗ Recall E = cB.
√
∗ Using c = 1/ 0 µ0 , it follows that
uE = uB
∗ Energy in an electromagnetic wave is shared equally between the
constituent electric and magnetic fields.
∗ If u = uB + uE , then
u = 0 E 2 ,
or
u − µ0 B 2 .
∗ Let S be the transport of energy per unit time (the power) across
a unit area (units W/m2 in SI units).
∗ Then, can show that
S=
uc∆tA
= uc,
∆tA
or
1
EB.
µ0
∗ The corresponding vector S is then
S=
S=
1
E × B,
µ0
or
S = c2 0 E × B.
∗ The magnitude of S is the power per unit area crossing a surface
whose normal is parrallel to S.
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∗ S is known as the Poynting vector.
∗ Now apply this to the case of a linearly polarized (directions of E
and B fields are fixed) plane wave travelling through free space in
the direction of k. Then
E = E 0 cos(k.r − ωt),
B = B 0 cos(k.r − ωt).
∗ Then
S = c2 0 E 0 B 0 cos2 (k.r − ωt).
∗ This is the instantaneous flow of energy per unit area per unit
time.
∗ This flow of energy per unit time per unit area is cyclic and cycles
from a maximuma to a minima.
– Irradiance
∗ The ”amount” of light illuminating a surface is called the irradiance, denoted by I, sometimes also called intensity.
∗ The irradiance I is the average energy per unit area per unit time
illuminating this surface - units W/m2 .
∗ The time - averaged value of the magnitude of the Poynting vector,
denoted by < S >T is a measure of I.
< S >T = c2 0 |E 0 × B 0 | < cos2 (k.r − ωt) > .
∗ Because < cos2 (x) >= 1/2,
< S >T =
c2 0
|E 0 × B 0 |,
2
or
c 2
E .
2 0
∗ So the irradiance is proportional to the square of the amplitude
of the electric field.
∗ Time rate of flow of radiant energy is the optical power, P or
radiant flux, units are watts.
∗ Divide the radiant flux incident on or exiting from a surface we
have the the radiant flux density W/m2 . This is irradiance or
exitance.
I =< S >T =
– The Inverse Square Law
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∗ Consider an isotropic point source in free space, emitting energy
equally in all directions - that is emitting spherical waves. Surround the source with two concentric sherical shells A, B with
radii r and 2r respectively.
∗ Since energy is conserved, the same amount energy flows through
shells A, B.
∗ But shell B has twice the radius which means 4 times the surface
area. So the energy is distributed over an area 4 times as large.
∗ That is the energy per unit area in shell B is 4 times as smaller.
∗ Another way to look at this is to note that the amplitude drops
off inversely with r.
∗ This is the inverse square law.
∗ Or multiply I by the surface area and taking the square root keads
to
rE0 (r) = constant.
∗ The amplitude drops of inversely with r and thus the irradiance
is inversely proportional to 1/r 2.
– Photons
∗ Light absorbed and emitted in tiny discrete bursts, in particles
known as ”photons”.
∗ In situations think of light as a wave, and sometimes as particles.
∗ One of the outstanding problems in in 1900 was what happens
when EM waves are in quilibrium with an isothermal chamber
(black box). All the EM radiation is absorbed and emitted by the
walls of the enclosure - none enters from outside. The goals was
to predict the spectrum of radiation emanating from the the black
body.
∗ Planck solved this problem with a description of blackbody radiation.
∗ He postulated that each one of the ”oscillators” lining the walls
of the chamber could absorb and emit radation only in discrete
amounts of energy propotional to the oscillator frequency, ν.
∗ Energy jolts were hν, where Planck’s constant, h = 6.626 ×
10−34 Js.
∗ At this time the wave picture of light was prevalent and Plank
hypothesized that only the oscillators were quantized.
∗ JJ Thompson, who discovered the electron, extended this idea and
suggested that electromagnetic waves might be radically different
from other waves - perhaps local concentrations of radiant energy
existed.
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∗ Concept of the photon was developed by Einstein’s work on the
photoelectric effect.
∗ A metal which is bathed in EM radiation emits electrons but this
emission cannot be described in light is a classical wave - only if
the electromagnetic field itslef is quantized.
∗ Each constituent photon has an energy given by
E = hν.
∗ Photons are stable, chargeless, massless elementary particles that
exist only at the speed c.
∗ What is known of photons comes from observing the results of
their being created or destroyed.
∗ Imagine a very dim source, surrounded at equal distances, by indentical photodetectors capable of measuring a minute amount of
light.
∗ If the emission, no matter how faint, is a continuous wave, all the
detectors should register each emitted pulse in coincidence.
∗ This does not happen - counts are registered by detctors independently, one at a time, in agreement with the idea that atoms emit
localized light quanta in random directions.
∗ When an atom emits light, it recoils in the opposite direction like
a pistol firing a bullet.
– A barrage of Photons
∗ Analyzing phenonemna involving the activity of large numbers of
particpants, statistical techniques are ooften needed. Two types
in use here:
∗ Bose-Einstein statistics: particles that are not subject to the Pauli
Exclusion princple - bosons.
∗ Fermi-Dirac statistics: particles that are subject to the Pauli Exclusion principle - fermions.
∗ Photons are bosons, electrons are fermions.
∗ Only one fermion can occupy any given state. Any number of
bosons can occupy the same state.
∗ When a very large number of photons occupy the same state, the
inherent granularity of the light beam essentially vanishes and
the electromagnetic field appears as the continuous medium of an
electromagnetic wave.
∗ Thus associate a monochromatic wave with a stream of photons
having a high population densitu, all progressing in the same state
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∗
∗
∗
∗
(with the same energy, same frequency, same momentum, same direction). Different monochromatic plane waves represent different
photon states.
Because electrons are fermions, large numbers of them cannot
cluster tightly in the same state and a monoenergetic beam of
electrons does not manifest itself on a macroscopic scale as a continuous classical wave.
A large number of photons can occupy the same state and when
they do, the inherent granularity of the light beam vanishes and
the electromagnetic field appears as the continuous medium of an
electromagnetic wave.
For a uniform monochromatic light beam of frequency ν, the quantity I/hν is the average number of photons impinging on a unit
area (normal to the beam) per unit time - the photon flux density.
Light delivers its energy in a staccato of impacts that are random
in space and time across the beam.
– Radiation Pressure and Momentum
∗ Waves and particles exert a pressure
∗ When an EM wave impinges on some material surface, it interacts
with the charges that constitute the bulk matter.
∗ Regardless of whether the wave is partially absorbed or reflected,
it exerts a force on those charges and hence on the surface.
∗ The wave itself carries momentum.
∗ The radiation pressure P equals the energy density of the wave.
∗ Then for a vacuum,
0
uE = E 2 ,
2
1 2
uB = B .
µ0
∗ Since P = u = uE + uB ,
P=
0 2
1 2
E +
B .
2
2µ0
∗ S, the Poynting vector refers to the transport of energy per unit
time across a unit area and pressure which is force/unit area is
energy density, then,
S(t)
.
P=
c
∗ Check this equation has units of force/area.
∗ This is the instantaneous pressure that would be exerted on a
perfectly absorbing surface by a normally indcident beam.
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∗ Taking a time average yields,
I
P= .
c
∗ This leads to photons carrying a momentum
p=
h
E
= .
c
λ
∗ Radiation pressure important in large, bright stars and sets a limit
(the Eddington limit) on the largest mass possible for a star.
– Radiation
∗ Though, EM radiation comes in wide range of wavelengths and
frequencies, Maxwell’s equations are independent of frequency therefore there will be a common source mechanism.
∗ EM radiation associated with nonuniformly moving charges.
∗ A stationary charge has a constant E field, no B field and hence
no radiation.
∗ A uniformly moving charge has both E and B fields but it does
not radiate.
∗ Travelling along with this uniformly moving charge, the current
would vanish, hence B would vanish etc.
∗ If a charge moves uniformly, it radiates.
– Linearly Accelerating charges
∗ Fig. 3.27, p. 59 shows the electric field lines for a charge accelerating to the right.
∗ Field lines are now curved and so there exists a trasnverse component of the electric field which propogates outward as pulse.
∗ At some point, this transverse electric field will be a function of
time and will therefore be accompanied by a magnetic field.
∗ Enerrgy is most strongly radiated perpendicular to the acceleration causing it.
∗ The energy thats radiated out is supplied by the agent responsible
for accelerating the charge.
– Synchrotron Radiation
∗ A free charged particle travelling on any sort of curved path is
accelerating and will radiate.
∗ Charged particles trapped in external magnetic fields will move in
circular or helical orbits and if their speeds are high enough - they
will emit synchrotron raidation eg. Crab Nebula.
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– Electric Dipole Radiation
∗ Consider an oscillating dipole: two charges, one plus, one minus,
vibrating to and fro along a straight line.
∗ The electrical dipole moment for a pair of opposite charges, one
positive, one negative is
p = qd,
where d is the distance between them.
∗ Since the charges oscillate and the distance between them changes,
the dipole moment is harmonically varying,
p = p0 cos(ωt).
∗ The E field changes dramatically as the distance between the
charges changes.
∗ Its complicated, but far from the dipole, a fixed wavelength is
established, E and B are transverse and mutually perpendicular
and in phase and it can be shown that the irradiance, given by
I(θ) =
p02 ω 4 sin2 θ
.
32π 2 c3 0 r 2
∗ This is again an inverse square law dependence on distance. Also,
the higher the frequence, the stronger the radiation.
– Emission of Light from Atoms
∗ Energy levels in atoms.
∗ Ground states and excited states.
∗ When atoms interact with light, they jump from initial to excited
states and since these energy states have well defined energies, the
amount of energy that can be absorbed by an atom from light is
quantized.
∗ The lifetime of the atom in this excited state is short (10−8 s and
the excited atom spntaneously decays back into a lower energy
state, losing the excitation energy along the way.
∗ This process results in the absorption of light at the resonant
frequency and the transmission or reflection of light at other frequencies - coloration of worl around us.
∗ Light energy carried by photons,
E = hν.
∗ The ν is associated with a specific frequency of the photon and the
atmoic transition between two particular states. These frequencies
are ”resonant frequencies” at which the atom efficiently absorbs
and emits energy.
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∗ Emission spectra of single atoms or low pressure gasses consist of
sharp well-defined frequencies characteristic of the atoms.
∗ Spectra of solids and liquids and more complicated gases where
the atoms may interact with one another broadens the spectra into
frequency bands - sometimes blurring thousands of frequencies
into a wide band.
– Optical Cooling
∗ Photons carry linear momentum which can be transferred to moving atoms and ions thus changing their motion.
∗ Its posible to ”slow” an atom to about zero speed after a considerable number of photon absorption and emission cycles.
∗ Optical or laser cooling.
– Light in Bulk Matter
∗
∗
∗
∗
So far considered EM waves travelling in free space.
Will be considering light in lenses, prisms, water etc.
Net effect is to change 0 and µ0 to and µ.
Phase speed in the medium is
v = 1/sqrtµ.
∗ Ratio of the speed of an electromagnetic wave in vacuum to that
in matter is known as the absolute index of refraction, n:
c
n= =
v
s
µ
.
0 µ0
∗ In terms of relative permittivity, n becomes
n=
q
KE KM
∗ Generally KM is close to 1, so setting KM = 1 gives Maxwell’s
relation,
q
n ≈ KE .
∗ Here KE iis the static dielectric constant.
∗ This only works for some substances because KE and thus n can
be frequency dependent.
∗ Dependenc of n on wavelength or color is called dispersion.
– Scattering and Absorption
∗ Consider the interaction of an incident EM wave with the array
oof atoms representing a dielectric material.
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∗ An atom can react in one of two ways depending on the incident
frequency.
∗ The atom can ”scatter” the light, redirecting it, without altering
it.
∗ If the photon’s energy matches that of one of the excited states,
the atom will absoorb the light, making a quantum jump to a
higher energy level.
∗ This ”extra energy” could be absorbed by other atoms through
collisions before the atom canspontaneously jump back down to
its lower energy state.
∗ ”Non-resonant or ground state scattering” occurs when the incoming photon’s energy is too low to kick the atom up to a higher
energy level.
∗ However it can still drive the electron cloud into oscillation.
∗ Once the elctron cloud starts to oscillate with respect to the positive nucleus, its an oscillating dipole and the system starts to
radiate.
∗ The resulting ”scattered” light goes of in some direction carrying
the same amount of energy as did the incident photon - elastic
scattering.
– Dispersion
∗ Dispersion corresponds to the phenomena whereby the index of
refraction of a medium is frequency dependent.
∗ All materiial media are dispersive - only a vacuum is not.
∗ When a dielectric is subjected to an applied electric field, as is the
case when it is illuminated by an EM wave, the internal charge
distribution is distorted.
∗ The external electric field sperates positive and negative charges
(each pair of which is a dipole).
∗ The resultant dipole moment per unit volume - the electric polarization, P is
( − 0 )E = P .
∗ Need to show why n2 = KE = /0 is dependent on the frequency
of the incident radiation.
∗ Think of the electrons as masses on the end of springs attached
to the nucleus.
∗ These electrons can undergo
Simple Harmonic Motion with a natq
ural frequency ω0 = kE /me , where me is the electron mass and
ke is a type of elastic constant.
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∗ When an EM wave of frequency ω strikes this system, each atom
can be thought of as a forced oscillator driven by the applied
electric field E(t), with a force,
FE = qe E(t) = qe E0 cos(ωt).
∗ The equation of motion of the electron, where x represents the
relative displacement between the negative electron cloud and the
positive nucleus,
qe E0 cos(ωt) − me ω02 x = me
d2 x
.
dt2
∗ This differential equation has a solution,
x(t) =
qe /me
E(t).
(ω02 − ω)
∗ GOing back to the dipole moment, P , this is equal to
P = qe xN,
where N is the contributin electrons per unit volume.
∗ Thus it can be shown that
= 0 +
qe2 N/me
.
(ω02 − ω 2)
∗ Since n2 = KE = /0 , we obtain the dispersion equation,
n2 (ω) = 1 +
1
Nqe2
( 2
).
0 me ω0 − ω 2
∗ Table 3.3, p. 70 shows the dispersion of Crown Glass as a function
of wavelength.
∗ A substance can have several frequencies like ω0 .
∗ A more detailed treatment can show that
n2 − 1
fj
Nqe2 X
=
.
2
2
n +1
30 me j ω0j − ω 2
∗ Colorless, transparent materials have their characteristic frequencies outside of the visible region of the spectrum - thats why they
are colorless and transparent.
∗ Glasses have effective natural frequencies above the visible in the
UV where they are opaque.
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∗ Normal dispersion - n gradually increasing with frequency.
∗ Regions surrounding the various ω0j are called absorption bands.
∗ In summary a material can be thought of as electron-oscillators
which have some natural frequency of vibrations corresponding to
the energy levels of the atom/molecule.
∗ When the incoming EM wave’s frequency is different to the natural
frequencies, there is little dissipative absorption. At resonance the
oscillator amplitudes are increased and energy is removed from the
incident EM beam at that frequency - leading to an absorption
peak or band. The material is opaque at this frequency.
∗ Read Section 3.6 on the electromagnetic - photon spectrum, p.
73-80.
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