Chapter 24

Electromagnetic Waves

Electromagnetic Waves, Introduction Electromagnetic (em) waves permeate

our environment EM waves can propagate through a

vacuum Much of the behavior of mechanical

wave models is similar for em waves Maxwell’s equations form the basis of

all electromagnetic phenomena

Conduction Current A conduction current is carried by charged

particles in a wire The magnetic field associated with this

current can be calculated using Ampère’s Law:

The line integral is over any closed path through which the conduction current passes

od I B s

Conduction Current, cont. Ampère’s Law in this

form is valid only if the conduction current is continuous in space

In the example, the conduction current passes through only S1

This leads to a contradiction in Ampère’s Law which needs to be resolved

Displacement Current Maxwell proposed the resolution to the

previous problem by introducing an additional term This term is the displacement current

The displacement current is defined as

Ed o

dI

dt

Displacement Current, cont. The changing electric field may be considered

as equivalent to a current For example, between the plates of a capacitor

This current can be considered as the continuation of the conduction current in a wire

This term is added to the current term in Ampère’s Law

Ampère’s Law, General The general form of Ampère’s Law is

also called the Ampère-Maxwell Law and states:

Magnetic fields are produced by both conduction currents and changing electric fields

( ) Eo d o o o

dd I I I

dt

B s

Ampère’s Law, General – Example The electric flux through

S2 is EA S2 is the gray circle A is the area of the

capacitor plates E is the electric field

between the plates If q is the charge on the

plates, then Id = dq/dt This is equal to the

conduction current through S1

James Clerk Maxwell 1831 – 1879 Developed the

electromagnetic theory of light

Developed the kinetic theory of gases

Explained the nature of color vision

Explained the nature of Saturn’s rings

Died of cancer

Maxwell’s Equations, Introduction In 1865, James Clerk Maxwell provided a

mathematical theory that showed a close relationship between all electric and magnetic phenomena

Maxwell’s equations also predicted the existence of electromagnetic waves that propagate through space

Einstein showed these equations are in agreement with the special theory of relativity

Maxwell’s Equations In his unified theory of

electromagnetism, Maxwell showed that electromagnetic waves are a natural consequence of the fundamental laws expressed in these four equations:

0o

B Eo o o

qd d

d dd d I

dt dt

E A B A

E s B s

Maxwell’s Equations, Details The equations, as shown, are for free space

No dielectric or magnetic material is present The equations have been seen in detail in

earlier chapters: Gauss’ Law (electric flux) Gauss’ Law for magnetism Faraday’s Law of induction Ampère’s Law, General form

Lorentz Force Once the electric and magnetic fields

are known at some point in space, the force of those fields on a particle of charge q can be calculated:

The force is called the Lorentz force

q q F E v B

Electromagnetic Waves In empty space, q = 0 and I = 0 Maxwell predicted the existence of

electromagnetic waves The electromagnetic waves consist of oscillating

electric and magnetic fields The changing fields induce each other which

maintains the propagation of the wave A changing electric field induces a magnetic field A changing magnetic field induces an electric field

Plane em Waves We will assume that the

vectors for the electric and magnetic fields in an em wave have a specific space-time behavior that is consistent with Maxwell’s equations

Assume an em wave that travels in the x direction with the electric field in the y direction and the magnetic field in the z direction

Plane em Waves, cont The x-direction is the direction of propagation Waves in which the electric and magnetic

fields are restricted to being parallel to a pair of perpendicular axes are said to be linearly polarized waves

We also assume that at any point in space, the magnitudes E and B of the fields depend upon x and t only

Rays A ray is a line along which the wave travels All the rays for the type of linearly polarized

waves that have been discussed are parallel The collection of waves is called a plane

wave A surface connecting points of equal phase

on all waves, called the wave front, is a geometric plane

Properties of EM Waves The solutions of Maxwell’s are wave-like, with

both E and B satisfying a wave equation Electromagnetic waves travel at the speed of

light

This comes from the solution of Maxwell’s equations

oo

1c

Properties of em Waves, 2 The components of the electric and

magnetic fields of plane electromagnetic waves are perpendicular to each other and perpendicular to the direction of propagation This can be summarized by saying that

electromagnetic waves are transverse waves

Properties of em Waves, 3 The magnitudes of the fields in empty

space are related by the expression

This also comes from the solution of the partial differentials obtained from Maxwell’s Equations

Electromagnetic waves obey the superposition principle

BEc

Derivation of Speed – Some Details From Maxwell’s equations applied to empty

space, the following partial derivatives can be found:

These are in the form of a general wave equation, with

Substituting the values for o and o gives c = 2.99792 x 108 m/s

2 2 2 2

2 2 2 2o o o o

E E B Band

x t x t

1 o ov c

E to B Ratio – Some Details The simplest solution to the partial differential

equations is a sinusoidal wave: E = Emax cos (kx – t)

B = Bmax cos (kx – t)

The angular wave number is k = 2 is the wavelength

The angular frequency is = 2 ƒ ƒ is the wave frequency

E to B Ratio – Details, cont The speed of the electromagnetic

wave is

Taking partial derivations also gives

2 ƒƒ

2c

k

max

max

E Ec

B k B

em Wave Representation This is a pictorial

representation, at one instant, of a sinusoidal, linearly polarized plane wave moving in the x direction

E and B vary sinusoidally with x

Doppler Effect for Light Light exhibits a Doppler effect

Remember, the Doppler effect is an apparent change in frequency due to the motion of an observer or the source

Since there is no medium required for light waves, only the relative speed, v, between the source and the observer can be identified

Doppler Effect, cont. The equation also depends on the laws of

relativity

v is the relative speed between the source and the observer

c is the speed of light ƒ’ is the apparent frequency of the light seen by

the observer ƒ is the frequency emitted by the source

ƒ ' ƒc v

c v

Doppler Effect, final For galaxies receding from the Earth v

is entered as a negative number Therefore, ƒ’<ƒ and the apparent

wavelength, ’, is greater than the actual wavelength

The light is shifted toward the red end of the spectrum

This is what is observed in the red shift

Heinrich Rudolf Hertz 1857 – 1894 Greatest discovery

was radio waves 1887

Showed the radio waves obeyed wave phenomena

Died of blood poisoning

Hertz’s Experiment An induction coil is

connected to a transmitter

The transmitter consists of two spherical electrodes separated by a narrow gap

Hertz’s Experiment, cont The coil provides short voltage surges to the

electrodes As the air in the gap is ionized, it becomes a

better conductor The discharge between the electrodes

exhibits an oscillatory behavior at a very high frequency

From a circuit viewpoint, this is equivalent to an LC circuit

Hertz’s Experiment, final Sparks were induced across the gap of the

receiving electrodes when the frequency of the receiver was adjusted to match that of the transmitter

In a series of other experiments, Hertz also showed that the radiation generated by this equipment exhibited wave properties Interference, diffraction, reflection, refraction and

polarization He also measured the speed of the radiation

Poynting Vector Electromagnetic waves carry energy As they propagate through space, they

can transfer that energy to objects in their path

The rate of flow of energy in an em wave is described by a vector called the Poynting vector

Poynting Vector, cont The Poynting Vector is

defined as

Its direction is the direction of propagation

This is time dependent Its magnitude varies in

time Its magnitude reaches a

maximum at the same instant as the fields

1

o S E B

Poynting Vector, final The magnitude of the vector represents

the rate at which energy flows through a unit surface area perpendicular to the direction of the wave propagation This is the power per unit area

The SI units of the Poynting vector are J/s.m2 = W/m2

Intensity The wave intensity, I, is the time

average of S (the Poynting vector) over one or more cycles

When the average is taken, the time average of cos2(kx-t) = ½ is involved

2 2max max max max

2 2 2avgo o o

E B E c BI S

c

Energy Density The energy density, u, is the energy per

unit volume For the electric field, uE= ½ oE2

For the magnetic field, uB = ½ oB2

Since B = E/c and oo1c 2

21

2 2B E oo

Bu u E

Energy Density, cont The instantaneous energy density

associated with the magnetic field of an em wave equals the instantaneous energy density associated with the electric field In a given volume, the energy is shared

equally by the two fields

Energy Density, final The total instantaneous energy density is the

sum of the energy densities associated with each field u =uE + uB = oE2 = B2 / o

When this is averaged over one or more cycles, the total average becomes uav = o (Eavg)2 = ½ oE2

max = B2max / 2o

In terms of I, I = Savg = c uavg The intensity of an em wave equals the average

energy density multiplied by the speed of light

Momentum Electromagnetic waves transport momentum

as well as energy As this momentum is absorbed by some

surface, pressure is exerted on the surface Assuming the wave transports a total energy

U to the surface in a time interval t, the total momentum is p = U / c for complete absorption

Pressure and Momentum Pressure, P, is defined as the force per unit

area

But the magnitude of the Poynting vector is (dU/dt)/A and so P = S / c For complete absorption An absorbing surface for which all the incident

energy is absorbed is called a black body

1 1F dp dU dtP

A A dt c A

Pressure and Momentum, cont For a perfectly reflecting surface,

p = 2 U / c and P = 2 S / c For a surface with a reflectivity somewhere

between a perfect reflector and a perfect absorber, the momentum delivered to the surface will be somewhere in between U/c and 2U/c

For direct sunlight, the radiation pressure is about 5 x 10-6 N/m2

Determining Radiation Pressure This is an apparatus for

measuring radiation pressure

In practice, the system is contained in a high vacuum

The pressure is determined by the angle through which the horizontal connecting rod rotates

Space Sailing A space-sailing craft includes a very large sail

that reflects light The motion of the spacecraft depends on the

pressure from light From the force exerted on the sail by the reflection

of light from the sun Studies concluded that sailing craft could

travel between planets in times similar to those for traditional rockets

The Spectrum of EM Waves Various types of electromagnetic waves

make up the em spectrum There is no sharp division between one

kind of em wave and the next All forms of the various types of

radiation are produced by the same phenomenon – accelerating charges

The EMSpectrum

Note the overlap between types of waves

Visible light is a small portion of the spectrum

Types are distinguished by frequency or wavelength

Notes on The EM Spectrum Radio Waves

Wavelengths of more than 104 m to about 0.1 m Used in radio and television communication

systems Microwaves

Wavelengths from about 0.3 m to 10-4 m Well suited for radar systems Microwave ovens are an application

Notes on the EM Spectrum, 2 Infrared waves

Wavelengths of about 10-3 m to 7 x 10-7 m Incorrectly called “heat waves” Produced by hot objects and molecules Readily absorbed by most materials

Visible light Part of the spectrum detected by the human

eye Most sensitive at about 5.5 x 10-7 m (yellow-

green)

More About Visible Light Different

wavelengths correspond to different colors

The range is from red ( ~7 x 10-7 m) to violet ( ~4 x 10-7 m)

Visible Light – Specific Wavelengths and Colors

Notes on the EM Spectrum, 3 Ultraviolet light

Covers about 4 x 10-7 m to 6 x10-10 m Sun is an important source of uv light Most uv light from the sun is absorbed in the

stratosphere by ozone X-rays

Wavelengths of about 10-8 m to 10-12 m Most common source is acceleration of high-

energy electrons striking a metal target Used as a diagnostic tool in medicine

Notes on the EM Spectrum, final Gamma rays

Wavelengths of about 10-10m to 10-14 m Emitted by radioactive nuclei Highly penetrating and cause serious

damage when absorbed by living tissue Looking at objects in different portions

of the spectrum can produce different information

Wavelengths and Information These are images of

the Crab Nebula They are (clockwise

from upper left) taken with x-rays visible light radio waves infrared waves

Polarization of Light Waves The electric and magnetic vectors associated

with an electromagnetic wave are perpendicular to each other and to the direction of wave propagation

Polarization is a property that specifies the directions of the electric and magnetic fields associated with an em wave

The direction of polarization is defined to be the direction in which the electric field is vibrating

Unpolarized Light, Example All directions of

vibration from a wave source are possible

The resultant em wave is a superposition of waves vibrating in many different directions

This is an unpolarized wave

The arrows show a few possible directions of the waves in the beam

Polarization of Light, cont A wave is said to be linearly

polarized if the resultant electric field vibrates in the same direction at all times at a particular point

The plane formed by the electric field and the direction of propagation is called the plane of polarization of the wave

Methods of Polarization It is possible to obtain a linearly

polarized beam from an unpolarized beam by removing all waves from the beam expect those whose electric field vectors oscillate in a single plane

The most common processes for accomplishing polarization of the beam is called selective absorption

Polarization by Selective Absorption

Uses a material that transmits waves whose electric field vectors in the plane parallel to a certain direction and absorbs waves whose electric field vectors are perpendicular to that direction

Selective Absorption, cont E. H. Land discovered a material that

polarizes light through selective absorption He called the material Polaroid The molecules readily absorb light whose

electric field vector is parallel to their lengths and allow light through whose electric field vector is perpendicular to their lengths

Selective Absorption, final It is common to refer to the direction

perpendicular to the molecular chains as the transmission axis

In an ideal polarizer, All light with the electric field parallel to the

transmission axis is transmitted All light with the electric field perpendicular

to the transmission axis is absorbed

Intensity of a Polarized Beam The intensity of the polarized beam

transmitted through the second polarizing sheet (the analyzer) varies as I = Io cos2 θ

Io is the intensity of the polarized wave incident on the analyzer

This is known as Malus’ Law and applies to any two polarizing materials whose transmission axes are at an angle of θ to each other

Intensity of a Polarized Beam, cont The intensity of the transmitted beam is

a maximum when the transmission axes are parallel = 0 or 180o

The intensity is zero when the transmission axes are perpendicular to each other This would cause complete absorption

Intensity of Polarized Light, Examples

On the left, the transmission axes are aligned and maximum intensity occurs

In the middle, the axes are at 45o to each other and less intensity occurs

On the right, the transmission axes are perpendicular and the light intensity is a minimum

Properties of Laser Light The light is coherent

The rays maintain a fixed phase relationship with one another

There is no destructive interference The light is monochromatic

It has a very small range of wavelengths The light has a small angle of divergence

The beam spreads out very little, even over long distances

Stimulated Emission Stimulated emission is required for laser

action to occur When an atom is in an excited state, an

incident photon can stimulate the electron to fall to the ground state and emit a photon

The first photon is not absorbed, so now there are two photons with the same energy traveling in the same direction

Stimulated Emission, Example

Stimulated Emission, Final The two photons (incident and emitted)

are in phase They can both stimulate other atoms to

emit photons in a chain of similar processes

The many photons produced are the source of the coherent light in the laser

Necessary Conditions for Stimulated Emission For the stimulated emission to occur,

there must be a buildup of photons in the system

The system must be in a state of population inversion More atoms must be in excited states than

in the ground state This insures there is more emission of

photons by excited atoms than absorption by ground state atoms

More Conditions The excited state of the system must be

a metastable state Its lifetime must be long compared to the

usually short lifetimes of excited states The energy of the metastable state is

indicated by E* In this case, the stimulated emission is

likely to occur before the spontaneous emission

Final Condition The emitted photons must be confined

They must stay in the system long enough to stimulate further emissions

In a laser, this is achieved by using mirrors at the ends of the system

One end is generally reflecting and the other end is slightly transparent to allow the beam to escape

Laser Schematic

The tube contains atoms The active medium

An external energy source is needed to “pump” the atoms to excited states

The mirrors confine the photons to the tube Mirror 2 is slightly transparent

Energy Levels, He-Ne Laser This is the energy level

diagram for the neon The neon atoms are

excited to state E3* Stimulated emission

occurs when the neon atoms make the transition to the E2 state

The result is the production of coherent light at 632.8 nm

Laser Applications Laser trapping Optical tweezers Laser cooling

Allows the formation of Bose-Einstein condensates