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Chapter 34 Electromagnetic The Goal of the Entire Course Maxwell’s Equations:

Maxwell’s Equations James Clerk Maxwell

•1831 – 1879 •Scottish theoretical physicist •Developed the electromagnetic theory of •His successful interpretation of the electromagnetic field resulted in the field equations that bear his name. •Also developed and explained – Kinetic theory of gases – Nature of Saturn’s rings – Color vision Start at 12:50 https://www.learner.org/vod/vod_window.html?pid=604 Correcting Ampere’s Law

Two surfaces S1 and S2 near the plate of a capacitor are bounded by the same path P. Ampere’s Law states that

But it is zero on S2 since there is no conduction current through it. This is a contradiction. Maxwell fixed it by introducing the displacement current:

Fig. 34-1, p. 984 Maxwell hypothesized that a changing creates an induced magnetic field. Induced Fields

. An increasing solenoid current causes an increasing magnetic field, which induces a circular electric field. . An increasing capacitor charge causes an increasing electric field, which induces a circular magnetic field.

Slide 34-50 Displacement Current d d(EA)d(q / ε) 1 dq E  0  dt dt dt ε0 dt

dq d  ε E dt0 dt

The displacement current is equal to the conduction current!!!

Bsd μ I  μ ε I  o o o d Maxwell’s Equations The First Unified Field Theory

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

q EABAdd    0 εo dd Edd s  BE B  s μ I  μ ε dto o o dt

QuickCheck 34.4

The electric field is increasing. Which is the induced magnetic field?

E. There’s no induced field in this case.

Slide 34-48 QuickCheck 34.4

The electric field is increasing. Which is the induced magnetic field?

E. There’s no induced field in this case.

Slide 34-49 We want to get EM Waves!! A sinusoidal electromagnetic moves in the positive x direction with a speed c.

Fig. 34-8, p. 990 Need Some1-C Vector Calculus

Put Maxwell’s Equations in Vector Form

Put Maxwell’s Equations in Vector Form 1st : Maxwell’s Equations in Free Space No charges or conduction currents: EABAdd 0   0 dd Edd s  BE B  s  με dtoo dt

Divergence: EB  0   0 BE Curl:  EB      με tt00

2nd: Take the Curl of Faraday’s and Ampere’s Laws Changing E Field Produces B field and visa versa

EB  xt Changing E Field Produces B field and visa versa

EB  xt 3rd: Derive EM Wave Equations From Maxwell’s equations applied to empty space, the following can be derived: 2EEBB  2  2  2 μ ε and μ ε x2o o  t 2  x 2 o o  t 2

These are one dimensional wave equations of the standard form:

With wave speed: 1 vc μεoo

4: Solutions are waves! The simplest solution to the partial differential equations is a sinusoidal wave:

E(x,t) = Emax cos (kx – ωt) B(x,t) = Bmax cos (kx – ωt)

E ω max c Bkmax

The angular wave number is k = 2π/λ λ is the wavelength ωπ2 ƒ The angular is ω = 2πƒ  λcƒ  ƒ is the wave frequency k 2πλ

The EM

•Note the overlap between types of waves •Visible light is a small portion of the spectrum. •Types are distinguished by frequency or wavelength Section 34.7 Energy of Light

106 eV 104 eV 12 eV 40eV KeV MeV

Energy to ionize atom or molecule: 10-1000eV Electromagnetic Waves Factoids

•Mechanical waves require the presence of a medium. •Electromagnetic waves can propagate through empty space. •Maxwell’s equations form the theoretical basis of all electromagnetic waves that propagate through space at the . •Hertz confirmed Maxwell’s prediction when he generated and detected electromagnetic waves in 1887. •Electromagnetic waves are generated by oscillating electric charges. – The waves radiated from the oscillating charges can be detected at great distances. •Electromagnetic waves carry energy and momentum. •Electromagnetic waves cover many . Summary: Properties of Electromagnetic Waves Any electromagnetic wave must satisfy four basic conditions: 1. The fields E and B and are perpendicular to the

direction of propagation vem.Thus an electromagnetic wave is a transverse wave. 2. E and B are perpendicular to each other in a manner

such that E × B is in the direction of vem.

3. The wave travels in vacuum at speed vem = c 4. E = cB at any point on the wave. Properties of Electromagnetic Waves The energy flow of an electromagnetic wave is described by the defined as

The magnitude of the Poynting vector is

The intensity of an electromagnetic wave whose electric

field amplitude is E0 is Energy Density •The energy density, u, is the energy per unit volume. 2 •For the electric field, uE= ½ εoE 2 •For the magnetic field, uB = ½ μoB •Since B = E/c and 2 1 2 B uuB E εE o  22μo

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. Intensity and Energy Density The intensity (brightness) of an EM wave equals the average energy density multiplied by the speed of light. It is also proportional to the Amplitude squared of the wave!

•The total instantaneous energy density is the sum of the energy densities associated with each field. 2 2 u =uE + uB = εoE = B / μo •When this is averaged over one or more cycles, the total average becomes 2 2 2 uavg = εo(E )avg = ½ εoE max = B max / 2μo

I = Savg = cuavg

2 2 2 uavg = εo(E )avg = ½ εoE max = B max / 2μo

E max  c Bmax

2 2 2 uavg = εo(E )avg = ½ εoE max = B max / 2μo QuickCheck 34.5

To double the intensity of an electromagnetic wave, you should increase the amplitude of the electric field by a factor of

A. 0.5. B. 0.707. C. 1.414. D. 2. E. 4.

Slide 34-61 QuickCheck 34.5

To double the intensity of an electromagnetic wave, you should increase the amplitude of the electric field by a factor of

A. 0.5. B. 0.707. C. 1.414. D. 2. E. 4

Slide 34-62 QuickCheck 34.6

An electromagnetic plane wave is coming toward you, out of the screen. At one instant, the electric field looks as shown. Which is the wave’s magnetic field at this instant?

E. The magnetic field is instantaneously zero.

© 2013 Pearson Education, Inc. Slide 34-63 QuickCheck 34.6

An electromagnetic plane wave is coming toward you, out of the screen. At one instant, the electric field looks as shown. Which is the wave’s magnetic field at this instant?

is in the direction of motion.

E. The magnetic field is instantaneously zero.

© 2013 Pearson Education, Inc. Slide 34-64 QuickCheck 34.7

In which direction is this electro-magnetic wave traveling?

A. Up. B. Down. C. Into the screen. D. Out of the screen. E. These are not allowable fields for an electromagnetic wave.

© 2013 Pearson Education, Inc. Slide 34-65 QuickCheck 34.7

In which direction is this electro-magnetic wave traveling?

A. Up. is in the direction of motion. B. Down. C. Into the screen. D. Out of the screen. E. These are not allowable fields for an electromagnetic wave.

© 2013 Pearson Education, Inc. Slide 34-66 It’s interesting to consider the force of an electromagnetic wave exerted on an object per unit , which is called the radiation pressure prad. The radiation pressure on an object that absorbs all the light is

IS av g where I is the intensity of the light wave.

For a perfectly reflecting surface, p = 2I/c=2S/c Radiation Pressure

. Electromagnetic waves transfer not only energy but also momentum. . Suppose we shine a beam of light on an object that completely absorbs the light energy. . The momentum transfer will exert an average radiation pressure on the surface: Artist’s conception of a future spacecraft powered by radiation pressure from where I is the intensity the sun. of the light wave.

Slide 34-70

What is the maximum radiation pressure exerted by sunlight in space (S = 1350 W/m2) on a highly polished silver surface?

For a perfectly reflecting surface, p = 2I/c=2S/c Accelerating Charges

All forms of the various types of EM waves are produced by the same phenomenon – accelerating charges.

Oscillating Charges & Microwave

Frequency of EM wave is the same as the frequency of oscillation. Heinrich Rudolf Hertz •1857 – 1894 •German physicist •First to generate and detect electromagnetic waves in a laboratory setting •The most important discoveries were in 1887. •He also showed other wave aspects of light. Section 34.2 Hertz’s Experiment •From a circuit viewpoint, this is equivalent to an LC circuit. •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 •He also measured the speed of the radiation. – It was close to the known value of the speed of light. Production of EM Waves by an

•This is a half-wave antenna. •Two conducting rods are connected to a source of alternating voltage. •The length of each rod is one-quarter of the wavelength of the radiation to be emitted. Antennas

. An antenna acts like an oscillating electric dipole, involving both moving charge and a current. . A self-sustaining electromagnetic wave is produced.

Slide 34-74 Far Fields: Radiation Fields

Fields detach from the dipole and propagate freely. Radio Waves

7. At a distance of 10 km from a radio transmitter, the amplitude of the E-field is 0.20 volts/meter. What is the total emitted by the radio transmitter?

8. What should be the height of a dipole antenna (of dimensions 1/4 wavelength) if it is to transmit 1200 kHz radiowaves? Production of EM Waves by an Antenna •The oscillator forces the charges to accelerate between the two rods. •The antenna can be approximated by an oscillating electric dipole. •The magnetic field lines form concentric circles around the antenna and are perpendicular to the electric field lines at all points. •The electric and magnetic fields are 90o out of phase at all times. •This dipole energy dies out quickly as you move away from the antenna. •The source of the radiation found far from the antenna is the continuous induction of an electric field by the time-varying magnetic field and the induction of a magnetic field by a time-varying electric field. •The electric and magnetic field produced in this manner are in phase with each other and vary as 1/r. •The result is the outward flow of energy at all times.

Section 34.6 Polarization of Light

Plane Polarized

Circular Polarized

Polarization upon Reflection Polarization of Light Waves

• The direction of polarization of each individual wave is defined to be the direction in which the electric field is vibrating • In this example, the direction of polarization is along the y-axis Polarization

. The plane of the electric field vector and the Poynting vector is called the plane of polarization. . The electric field in the figure below oscillates vertically, so this wave is vertically polarized.

Slide 34-75 Polarization

. The electric field in the figure below is horizontally polarized.

. Most natural sources of light are unpolarized, emitting waves whose electric fields oscillate randomly with all possible orientations.

Slide 34-76 Unpolarized Light

• 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 Plane Polarized Light Circularly Polarized EM Wave Elliptically Polarized EM Wave Polarization of Light

• A wave is said to be linearly polarized if the resultant electric field E vibrates in the same direction at all times at a particular point • The plane formed by E 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 except those whose electric field vectors oscillate in a single plane • Processes for accomplishing this include – selective absorption – reflection – double refraction – scattering Polarization by Selective Absorption

• The most common technique for polarizing light • Uses a material that transmits waves whose electric field vectors lie in the plane parallel to a certain direction and absorbs waves whose electric field vectors are perpendicular to that direction Polarization

The most common way of artificially generating polarized visible light is to send unpolarized light through a polarizing filter.

Slide 34-77 Malus’s Law

. Suppose polarized light of

intensity I0 approaches a polarizing filter. . The component of the incident electric field that is polarized parallel to the axis is transmitted:

. The transmitted intensity depends on the square of the electric field amplitude:

Slide 34-78 Polarizers and Analyzers

. Malus’s law can be demonstrated with two polarizing filters. . The first, called the polarizer, is used to produce

polarized light of intensity I0. . The , called the analyzer, is rotated by angle  relative to the polarizer.

Slide 34-81 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 Malus’ Law

2 II 0 cos  2 Polarization II 0 cos 

The intensity of unpolarized light passing through a polarizer will be reduced my ½, because the average value of the cosine squared term over all directions is ½. Polarizing Filters

. The transmission of the analyzer is (ideally) 100% when   0, and steadily decreases to zero when   90. . Two polarizing filters with perpendicular axes, called crossed polarizers, block all the light. . If the incident light on a polarizing filter is unpolarized, half the intensity is transmitted:

Slide 34-82 If the incident beam is linearly polarized along the vertical direction and has an intensity of 50 W/m2 and the angle is 30 degrees, which set up transmits more light? Determine the average intensity of the transmitted beam for both setups. If the light is unpolarized, which set up transmits the most light? QuickCheck 34.8

A vertically polarized light wave of intensity 1000 mW/m2 is coming toward you, out of the screen. After passing through this polarizing filter, the wave’s intensity is

A. 707 mW/m2. B. 500 mW/m2. C. 333 mW/m2. D. 250 mW/m2. E. 0 mW/m2.

Slide 34-79 QuickCheck 34.8

A vertically polarized light wave of intensity 1000 mW/m2 is coming toward you, out of the screen. After passing through this polarizing filter, the wave’s intensity is

A. 707 mW/m2. 2 2 B. 500 mW/m . I  I0 cos  C. 333 mW/m2. D. 250 mW/m2. E. 0 mW/m2.

Slide 34-80 Polarization Problem

Unpolarized light is passed through three successive Polaroid filters, each with its transmission axis at 45 to the preceding filter. What percentage of light gets through? a. 0% b. 12.5% c. 25% d. 50% e. 33% Polarization by Reflection

• When an unpolarized light beam is reflected from a surface, the reflected light may be – Completely polarized – Partially polarized – Unpolarized • It depends on the angle of incidence – If the angle is 0°, the reflected beam is unpolarized – For other angles, there is some degree of polarization – For one particular angle, the beam is completely polarized Polarization by Reflection, Partially Polarized • Unpolarized light is incident on a reflecting surface • The reflected beam is partially polarized • The refracted beam is partially polarized Polarization by Reflection, Completely Polarized • Unpolarized light is incident on a reflecting surface • The reflected beam is completely polarized • The refracted beam is perpendicular to the reflected beam • The angle of incidence is Brewster’s angle Polarizing Sunglasses

. Glare—the reflection of the sun and the skylight from roads and other horizontal surfaces—has a strong horizontal polarization. . This light is almost completely blocked by a vertical polarizing filter. . Vertically polarizing sunglasses can “cut glare” without affecting the main scene you wish to see.

Slide 34-85 Polarization by Reflection

• The angle of incidence for which the reflected beam is completely polarized is called the polarizing

angle, θp • Brewster’s law relates the polarizing angle to the index of refraction for the material

sin θp n tan θp cos θp

• θp may also be called Brewster’s angle Brewster’s Angle

sin θp n tan θp cos θp Sunlight reflected from a smooth ice surface is completely polarized. Determine the angle of incidence. (nice = 1.31.) a.52.6 b.25.6 c. 65.2 d.56.2 e. 49.8 Polarization by Double Refraction: Birefringence • In certain crystalline structures, the speed of light is not the same in all directions • Such materials are characterized by two indices of refraction • They are often called double-refracting or birefringent materials Optical Stress Analysis

• Some materials become birefringent when stressed • When a material is stressed, a series of light and dark bands is observed – The light bands correspond to of greatest stress • Optical stress analysis uses plastic models to test for regions of potential weaknesses Polarization by Scattering • The horizontal part of the electric field vector in the incident wave causes the charges to vibrate horizontally • The vertical part of the vector simultaneously causes them to vibrate vertically • If the observer looks straight up, he sees light that is completely polarized in the horizontal direction • Short wavelengths (blue) are scattered more efficiently than long wavelengths (red) • When sunlight is scattered by gas molecules in the air, the blue is scattered more intensely than the red • When you look up, you see blue • At sunrise or sunset, much of the blue is scattered away, leaving the light at the red end of the spectrum Optical Activity

• Certain materials display the property of optical activity – A material is said to be optically active if it rotates the plane of polarization of any light transmitted through it – Molecular asymmetry determines whether a material is optically active QuickCheck 34.9

Unpolarized light, traveling in the direction shown, is incident on polarizer 1. Does any light emerge from polarizer 3?

A. Yes. B. No.

Slide 34-83 QuickCheck 34.9

Unpolarized light, traveling in the direction shown, is incident on polarizer 1. Does any light emerge from polarizer 3?

A. Yes. B. No.

Slide 34-84 EM Spectrum cf  Increasing Energy Accelerating Charges

All forms of the various types of EM waves are produced by the same phenomenon – accelerating charges. Oscillating Charges Radio & Microwave

Frequency of EM wave is the same as the frequency of oscillation. A Problem with Electrodynamics The force on a moving charge depends on the Frame. Charge Rest Frame Wire Rest Frame (moving with charge) (moving with wire)

F  0 F qvBsin

Einstein realized this inconsistency and could have chosen either: •Keep Maxwell's Laws of Electromagnetism, and abandon Galileo's Spacetime •or, keep Galileo's Space-time, and abandon the Maxwell Laws. On the Electrodynamics of Moving Bodies 1905 Lorentz Contraction in Wire Moving charges in the wire cause a Lorentz contraction in the distance between the moving charges in the wire so that from the rest frame of the charge outside the wire (or at rest in the wire) the moving charges bunch up and thus give a net charge to the wire. Eisntein Saves Maxwell! The force on a moving charge does NOT depend on the Frame. Charge Rest Frame Wire Rest Frame (moving with charge) (moving with wire)

F qvBsin

2 F kq12 q/ r

From the rest frame of the charge, the wire is charged and it feels a Coulomb force. The forces in each frame are equal though they are due to different causes! E or B? It Depends on Your Perspective Einstein doesn’t like it: The Force Depends on Your Perspective

Whether a force is “electric” or “magnetic” depends on the motion of the reference frame relative to the sources. The Transformation of Electric and Magnetic Fields

. The Galilean field transformation equations are:

where is the of reference frame B relative to frame A. . The fields are measured at the same point in space by experimenters at rest in each reference frame.

. These equations are only valid if vBA  c.

Slide 32 https://www.youtube.com/watch?v=1TKSfAkWWN0 QuickCheck 34.3

Experimenters on earth have created the magnetic field shown. A rocket flies through the field, from right to left. Which are the field (or fields) in the rocket’s reference frame?

© 2013 Pearson Education, Inc. Slide 34-35 QuickCheck 34.3

Experimenters on earth have created the magnetic field shown. A rocket flies through the field, from right to left. Which are the field (or fields) in the rocket’s reference frame?

© 2013 Pearson Education, Inc. Slide 34-36 Einstein’s Principle of Relativity

• Maxwell’s equations are true in all inertial reference frames. • Maxwell’s equations predict that electromagnetic waves, including light, travel at speed c = 3.00 × 108 m/s =300,000km/s = 300m/µs . • Therefore, light travels at speed c in all inertial reference frames. Every experiment has found that light travels at 3.00 × 108 m/s in every inertial reference frame, regardless of how the reference frames are moving with respect to each other. Special Theory: Inertial Frames: Frames do not accelerate relative to eachother; Flat Spacetime – No Gravity.

General Theory: Noninertial Frames: Frames accelerate: Curved Spacetime, Gravity & acceleration.