1. Electromagnetic Waves
Chapter 24
Electromagnetic Waves

2. 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

3. 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

4. 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

5. 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

6. 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

7. 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

8. 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

9. 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

10. 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

11. 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:

12. 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

13. 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

14. 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

15. 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

16. 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

17. 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

18. 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

19. 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

20. 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

21. 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 mo and eo gives c = x 108 m/s

22. E to B Ratio – Some Details
The simplest solution to the partial differential equations is a sinusoidal wave:
E = Emax cos (kx – wt)
B = Bmax cos (kx – wt)
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

23. E to B Ratio – Details, cont
The speed of the electromagnetic wave is
Taking partial derivations also gives

24. 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

25. 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

26. 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

27. 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

28. Heinrich Rudolf Hertz 1857 – 1894 Greatest discovery was radio waves
1887
Showed the radio waves obeyed wave phenomena
Died of blood poisoning

29. Hertz’s Experiment An induction coil is connected to a transmitter
The transmitter consists of two spherical electrodes separated by a narrow gap

30. 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

31. 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

32. 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

33. 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

34. 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

35. 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-wt) = ½ is involved

36. Energy Density The energy density, u, is the energy per unit volume
For the electric field, uE= ½ eoE2
For the magnetic field, uB = ½ moB2
Since B = E/c and

37. 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

38. Energy Density, final
The total instantaneous energy density is the sum of the energy densities associated with each field
u =uE + uB = eoE2 = B2 / mo
When this is averaged over one or more cycles, the total average becomes
uav = eo (Eavg)2 = ½ eoE2max = B2max / 2mo
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

39. 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 Dt, the total momentum is p = U / c for complete absorption

40. 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

41. 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

42. 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

43. 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

44. 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

45. 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

46. 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 m
Well suited for radar systems
Microwave ovens are an application

47. 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)

48. More About Visible Light
Different wavelengths correspond to different colors
The range is from red (l ~7 x 10-7 m) to violet (l ~4 x 10-7 m)

49. Visible Light – Specific Wavelengths and Colors

50. 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 m
Most common source is acceleration of high-energy electrons striking a metal target
Used as a diagnostic tool in medicine

51. Notes on the EM Spectrum, final
Gamma rays
Wavelengths of about 10-10m to 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

52. 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

53. 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

54. 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

55. 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

56. 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

57. 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

58. 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

59. 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

60. 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

61. Intensity of a Polarized Beam, cont
The intensity of the transmitted beam is a maximum when the transmission axes are parallel
q = 0 or 180o
The intensity is zero when the transmission axes are perpendicular to each other
This would cause complete absorption

62. 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

63. 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

64. 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

65. Stimulated Emission, Example

66. 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

67. 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

68. 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

69. 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

70. 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

71. 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 nm

72. Laser Applications Laser trapping Optical tweezers Laser cooling
Allows the formation of Bose-Einstein condensates