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Chapter 24

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Chapter 24

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  1. 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 = 2.99792 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 EMSpectrum • 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 10-4 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 10-12 m • Most common source is acceleration of high-energy electrons striking a metal target • Used as a diagnostic tool in medicine