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**Electromagnetic Waves**

Chapter 24 Electromagnetic Waves

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**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

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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

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**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

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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

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**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

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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

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**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

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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

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**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

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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:

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**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

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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

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**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

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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

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**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

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**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

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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

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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

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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

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**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

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**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

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**E to B Ratio – Details, cont**

The speed of the electromagnetic wave is Taking partial derivations also gives

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**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

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**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

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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

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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

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**Heinrich Rudolf Hertz 1857 – 1894 Greatest discovery was radio waves**

1887 Showed the radio waves obeyed wave phenomena Died of blood poisoning

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**Hertz’s Experiment An induction coil is connected to a transmitter**

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

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**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

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**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

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**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

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**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

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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

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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

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**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

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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

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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

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**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

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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

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**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

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**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

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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

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**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

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**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

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**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

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**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)

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**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)

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**Visible Light – Specific Wavelengths and Colors**

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**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

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**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

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**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

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**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

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**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

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**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

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**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

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**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

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**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

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**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

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**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

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**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

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**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

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**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

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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

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**Stimulated Emission, Example**

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**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

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**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

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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

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**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

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**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

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**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

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**Laser Applications Laser trapping Optical tweezers Laser cooling**

Allows the formation of Bose-Einstein condensates

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