Chapter 33 Electromagnetic Waves Today s information age is based almost entirely on the physics of electromagnetic waves. The connection between electric and magnetic fields to produce light is own of the greatest achievements produced by physics, and electromagnetic waves are at the core of many fields in science and engineering. In this chapter we introduce fundamental concepts and explore the properties of electromagnetic waves. 33-1
Maxwell s Rainbow The wavelength/frequency range in which electromagnetic (EM) waves (light) are visible is only a tiny fraction of the entire electromagnetic spectrum Fig. 33-2 Fig. 33-1 33-2
The Travelling Electromagnetic (EM) Wave, Qualitatively An LC oscillator causes currents to flow sinusoidally, which in turn produces oscillating electric and magnetic fields, which then propagate through space as EM waves Next slide Fig. 33-3 Oscillation Frequency: ω = 1 LC 33-3
The Travelling Electromagnetic (EM) Wave, Qualitatively EM fields at P looking back toward LC oscillator r r 1. Electric E and magnetic B fields always perpendicular to direction in which wave is travelling transverse wave (Ch. 16) r r 2. E always perpendicular to B r r 3. E B always gives direction of wave travel r r 4. E and B vary sinusoidally (in time and space) and are in phase (in step) with each other Fig. 33-4 33-4
Mathematical Description of Travelling EM Waves Electric Field: E = E ( kx ωt) m sin Magnetic Field: B = B ( kx ωt) m sin Wave Speed: c = 1 µ ε All EM waves travel a c in vacuum EM Wave Simulation Wavenumber: k = Angular frequency: 2π λ ω = 2π τ Vacuum Permittivity: ε Fig. 33-5 Amplitude Ratio: Vacuum Permeability: µ Em c B = E( t) Magnitude Ratio: m B() t = c 33-5
A Most Curious Wave Unlike all the waves discussed in Chs. 16 and 17, EM waves require no medium through/along which to travel. EM waves can travel through empty space (vacuum)! Speed of light is independent of speed of observer! You could be heading toward a light beam at the speed of light, but you would still measure c as the speed of the beam! c = 299 792 458 m/s 33-6
The Travelling EM Wave, Quantitatively Induced Electric Field Changing magnetic fields produce electric fields, Faraday s law of induction dφ B Eds r r Ñ g = dt E r r Ñ g d s = ( E + de) Eh = h de Φ B = ( B)( hdx) db de db hde= hdx = dt dx dt Fig. 33-6 E B = x t E B = kemcos kx t and = Bmcos x t kx t Em kemcos( kx ωt) = ωbmcos( kx ωt) = c B ( ω ) ω ( ω ) m 33-7
The Travelling EM Wave, Quantitatively Induced Magnetic Field Changing electric fields produce magnetic fields, Maxwell s law of induction Fig. 33-7 dφ E Bds r r Ñ g = µε ur dt B ds r Ñ g = ( B+ db) Bh= h db dφ E de Φ E = ( E)( h dx) = h dx dt dt db hdb= µε hdx dt B E = µε x t ( ω ) µεω cos( ω ) kb cos kx t = E kx t m ( ) Em 1 1 1 = = = c c= B µε ω k µε c µ ε m m 33-8
Energy Transport and the Poynting Vector EM waves carry energy. The rate of energy transport in an EM wave is characterized by the Poynting vector S r Poynting Vector: r 1 r r S = E B µ The magnitude of S is related to the rate at which energy is transported by a wave across a unit area at any instant (inst). The unit for S is (W/m 2 ) energy/time power S = = area inst area inst The direction of S r at any point gives the wave's travel direction and the direction of energy transport at that point 33-9
Energy Transport and the Poynting Vector r r r r Since E B E B = EB Instantaneous 1 S = 1 E energy flow rate: S = EB and since B = µ c Note that S is a function of time. The time-averaged value for S, S avg is also called the intensity I of the wave. I energy/time power = Savg = = area area 1 1 = = = sin m ( ω ) 2 2 2 I Savg E E kx t cµ avg cµ avg E 1 m 2 E rms = I = E rms 2 cµ 2 2 1 2 1 2 1 1 B ue = εe = ε( cb) = ε B = = u 2 2 2 µε 2µ avg avg cµ B EB 33-1
Variation of Intensity with Distance Consider a point source S that is emitting EM waves isotropically (equally in all directions) at a rate P S. Assume energy of waves is conserved as they spread from source. How does the intesnity (power/area) change with distance r? Fig. 33-8 I power P = = S area 4π r 2 33-11
Radiation Pressure EM waves have linear momentum as well as energy light can exert pressure S r incident p Total absorption: p = U c p r = I c S r reflected S r incident p F = t power I = = area U t = A p energy/time area Total reflection Back along path: r p = 2 U c U = IA t IA F = (total absorption) c 2IA F = (total reflection back along path) c F p = A Radiation Pressure p r = 2 I c 33-12
Polarization The polarization of light is describes how the electric field in the EM wave oscillates. Vertically plane-polarized (or linearly polarized) Fig. 33-1 33-13
Polarized Light Unpolarized or randomly polarized light has its instantaneous polarization direction vary randomly with time Fig. 33-11 One can produce unpolarized light by the addition (superposition) of two perpendicularly polarized waves with randomly varying amplitudes. If the two perpendicularly polarized waves have fixed amplitudes and phases, one can produce different polarizations such as circularly or elliptically polarized light. Polarized Light Simulation 33-14
Polarizing Sheet I I Fig. 33-12 Only electric field component along polarizing direction of polarizing sheet is passed (transmitted), the perpendicular component is blocked (absorbed) 33-15
Intensity of Transmitted Polarized Light Intensity of transmitted light, unpolarized incident light: I = 1 2 I Since only the component of the incident electric field E parallel to the polarizing axis is transmitted Fig. 33-13 E = E = Ecosθ transmitted Intensity of transmitted light, polarized incident light: y 2 I = I cos θ For unpolarized light, θ varies randomly in time ( 2 ) ( 2 cos θ cos θ) I = I = I = I avg avg 1 2 33-16
Reflection and Refraction Although light waves spread as they move from a source, often we can approximate its travel as being a straight line geometrical optics What happens when a narrow beam of light encounters a glass surface? Law of Reflection θ ' = θ Reflection: 1 1 Fig. 33-17 Snell s Law n sinθ sinθ Refraction: 2 2 1 1 sinθ n sinθ 1 2 1 n2 n is the index of refraction of the material = = n 33-17
Sound Waves For light going from n 1 to n 2 n 2 = n 1 θ 2 = θ 1 n 2 > n 1 θ 2 <θ 1, light bent towards normal n 2 < n 1 θ 2 > θ 1, light bent away from normal Fig. 33-18 33-18
Chromatic Dispersion The index of refraction n encountered by light in any medium except vacuum depends on the wavelength of the light. So if light consisting of different wavelengths enters a material, the different wavelengths will be refracted differently chromatic dispersion Fig. 33-19 Fig. 33-2 n 2blue >n 2red Chromatic dispersion can be good (e.g., used to analyze wavelength composition of light) or bad (e.g., chromatic aberration in lenses) 33-19
Chromatic Dispersion Chromatic dispersion can be good (e.g., used to analyze wavelength composition of light) prism Fig. 33-21 or bad (e.g., chromatic aberration in lenses) lens 33-2
Rainbows Sunlight consists of all visible colors and water is dispersive, so when sunlight is refracted as it enters water droplets, is reflected off the back surface, and again is refracted as it exits the water drops, the range of angles for the exiting ray will depend on the color of the ray. Since blue is refracted more strongly than red, only droplets that are closer the the rainbow center (A) will refract/reflect blue light to the observer (O). Droplets at larger angles will still refract/reflect red light to the observer. What happens for rays that reflect twice off the back surfaces of the droplets? Fig. 33-22 33-21
Total Internal Reflection For light that travels from a medium with a larger index of refraction to a medium with a smaller medium of refraction n 1 >n 1 θ 2 >θ 1, as θ 1 increases, θ 2 will reach 9 o (the largest possible angle for refraction) before θ 1 does. n 2 n sinθ = n sin 9 = n 1 c 2 2 Critical Angle: θ c = sin n n 1 2 1 n 1 Fig. 33-24 Total internal reflection can be used, for example, to guide/contain light along an optical fiber When θ 2 > θ c no light is refracted (Snell s Law does not have a solution!) so no light is transmitted Total Internal Reflection 33-22
Fig. 33-27 In which direction does light reflecting off a lake tend to be polarized? Polarization by Reflection When the refracted ray is perpendicular to the reflected ray, the electric field parallel to the page (plane of incidence) in the medium does not produce a reflected ray since there is no component of that field perpendicular to the reflected ray (EM waves are transverse). Applications 1. Perfect window: since parallel polarization is not reflected, all of it is transmitted 2. Polarizer: only the perpendicular component is reflected, so one can select only this component of the incident polarization Brewster s Law θb + θr = 9 n sinθ = n sinθ 1 B 2 r ( ) n sinθ = n sin 9 θ = n cosθ 1 n2 Brewster Angle: θ B = tan n 1 B 2 B 2 B 1 33-23