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

particles in a wire is carried by charged  The magnetic field associated with this current can be calculated using Amp ère’s Law: 

B



d

s

 

o I

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

Conduction Current, cont.

   Amp ère’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 S 1 This leads to a contradiction in Amp ère’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

I d

 

o d



dt E

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 Amp ère’s Law

Amp ère’s Law, General

  The general form of Amp ère’s Law is also called the Amp ère-Maxwell Law and states: 

B



d

s

 

o

(

I



I d

)  

o I

  

o o d



dt

Magnetic fields are produced by both

E

conduction currents and changing electric fields

Amp ère’s Law, General – Example

   The electric flux through S 2 is EA  S 2 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 I d = dq/dt This is equal to the conduction current through S 1

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:  

E



d

A E



d

s

  

q



o d



B dt



B



d

A

 0 

B



d

s

 

o I

  

o o d



E dt

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 Amp ère’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:

F



q

E



q

  The force is called the Lorentz force

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 c  1  o  o  This comes from the solution of Maxwell’s equations

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 c  E B  This also comes from the solution of the partial differentials obtained from Maxwell’s Equations  Electromagnetic waves obey the superposition principle

Derivation of Speed – Some Details

   From Maxwell’s equations applied to empty space, the following partial derivatives can be found:  2

E



x

2   

o o

 2

E



t

2

and

 2

B



x

2   

o o

 2

B



t

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

v

1  

o o

Substituting the values for  o c = 2.99792 x 10 8 m/s and  o gives

E to B Ratio – Some Details

   The simplest solution to the partial differential equations is a sinusoidal wave: 

E

=

E

max cos (kx – w t) 

B

=

B

max cos (kx – w t) The angular wave number is k = 2 p / l  l is the wavelength The angular frequency is w = 2 p ƒ  ƒ is the wave frequency

E to B Ratio – Details, cont

 The speed of the electromagnetic wave is w

k

 2 p l  l ƒ 

c

 Taking partial derivations also gives

E

max

B

max  w

k E B



c

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

c



v

    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

Doppler Effect, final

 For galaxies receding from the Earth v is entered as a negative number  Therefore, ƒ’<ƒ and the apparent wavelength, l ’, 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

S

 1 

o

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

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 .

m 2 = W/m 2

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 cos 2 (kx w t) = ½ is involved

I



S avg



E

max 2

B

max 

o



E

2 max 2 

o c



c B

2 max 2 

o

Energy Density

    The energy density,

u

, is the energy per unit volume For the electric field,

u

E = ½  o E 2 For the magnetic field,

u

B Since B = E/c and c  1 = ½  o B 2  o  o

u B



u E

 1 

o E

2 2 

B

2 2 

o

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

sum of the energy densities associated with each field instantaneous energy density is the 

u

=

u

E +

u

B =  o E 2 = B 2 /  o When this is averaged over one or more cycles, the total average becomes 

u av

=  o (E avg ) 2 = ½  o E 2 max = B 2 max / 2  o In terms of I, I = S avg  = c u avg 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 D t, the total momentum is p = U / c for complete absorption

Pressure and Momentum

  Pressure, P, is defined as the force per unit area

P



F

1

dp

1

dU dt A



A dt



c A

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

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/m 2

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 EM Spectrum

   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 10 4 m to about 0.1 m  Used in radio and television communication systems  Microwaves  Wavelengths from about 0.3 m to 10 -4  Well suited for radar systems  Microwave ovens are an application m

Notes on the EM Spectrum, 2

  Infrared waves   Wavelengths of about 10 -3 m to 7 x 10 -7 Incorrectly called “heat waves” m   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 green) m (yellow-

More About Visible Light

 Different wavelengths correspond to different colors  The range is from red ( l ~7 x 10 -7 to violet ( l m) ~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 -10 m 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

a particular point at The plane formed by the electric field and the direction of propagation is called the

plane of polarization

wave of the

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 = I o  cos 2 θ I o 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  q = 0 or 180 o  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 45 o less intensity occurs to each other and 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 E 3 * Stimulated emission occurs when the neon atoms make the transition to the E 2 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