Chapter 34 Electromagnetic Waves 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 light •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 electric field 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!!! Bsd 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 EABAdd 0 εo dd Edd 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 wave 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: EABAdd 0 0 dd Edd 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 frequency is ω = 2πƒ λcƒ ƒ is the wave frequency k 2πλ 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 Section 34.7 Energy of Light 106 eV 104 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 speed of light. •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 frequencies. 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 Poynting vector 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 Radiation Pressure It’s interesting to consider the force of an electromagnetic wave exerted on an object per unit area, 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 Radiation Pressure 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 Radio & 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 polarization •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 Antenna •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.