Physics 208, Lecture 22 Today’s Topics Electromagnetic Waves (EM Waves) Review: Waves and Wave Equation Maxwell’s Equations Propagation of E and B Energy Carried by EM Wave, Poynting Vector Momentum Carried by EM Wave Spectrum of EM wave. 1Maxwell Equations € E • dA =qε0∫ € E • d = −dΦBdt∫€ F = qE + qv × B € B• d =µ0I +ε0µ0dΦEdt∫€ B• dA = 0∫Gauss’s Law/ Coulomb’s Law Faraday’s Law Gauss’s Law of Magnetism, no magnetic charge Ampere Maxwell Law Also, Lorentz force Law These are the foundations of electromagnetism 2EM Fields in Space Maxwell equations when there is no charge and current: ∫=• 0AE d € E • d = −dΦBdt∫ € B• d =ε0µ0dΦEdt∫∫=• 0AB dtBxEzy∂∂−=∂∂tExByz∂∂−=∂∂00εµdifferential forms: (single polarization) x y z 220022tExEyy∂∂=∂∂εµ220022tBxBzz∂∂=∂∂εµE B 3Linear Wave Equation Linear wave equation 222221tyvxy∂∂=∂∂certain physical quantity Wave speed Sinusoidal wave y = Asin2πλx − 2πft +ϕ⎛⎝⎜⎞⎠⎟A:Amplitude f: frequency φ:Phase General wave: sum of sinusoidal waves v=λf k=2π/λ$ω=2πf λ:wavelength 4Electromagnetic Waves EM wave equations: Plane wave solutions: E= Emaxsin(kx-ωt+φ) B= Bmaxsin(kx-ωt+φ) Properties: No medium is necessary. E and B are normal to each other E and B are in phase Direction of wave is normal to both E and B (EM waves are transverse waves) Speed of EM wave: E/B = Emax/Bmax=c Transverse wave: two polarizations possible same φ, set to be 0 smc /109972.21800×==εµx y z E B c 5The EM Wave x y z E B c Two polarizations possible 6Wavelength and Frequency Because of the wave equation the wavelength of and frequency of a EM wave in vacuum are related by: Example: Determine the wavelength of an EM wave of frequency 50 MHz in free space € λf = c = 3⋅108m / s€ λ=cf=3⋅108m / s50 MHz=3⋅108m / s5 ⋅107s−1= 6m7Energy Carried By EM Waves Recall: energy densities uE= ½ ε0E2, uB= ½ B2/µ0 For a EM wave, at any time/location, uE = ½ ε0E2 = ½ B2/µ0 =uB (using E/B=c) In an electromagnetic wave, the energies carried by electric field and magnetic field are always the same. Total energy stored (per unit of volume): u=uE+uB = ε0E2 = B2/µ0 Power transmitted per unit of area is equal to uc in the direction of wave Averaging over time: uav= ½ ε0Emax2 = ½ Bmax2/µ0 , uavc = I (intensity) x y z E B 8The Poynting Vector The rate of flow of energy in an em wave is described by a vector, S , called the Poynting vector 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 E and B € S ≡1µ0 E × B Power per unit of area € I = Sav9Momentum Carried By EM Waves EM waves: momentum = energy/c Radiation Pressure (P): A ΔX=cΔt S 100% absorption Δp = p P= S/c 100% reflection Δp = 2p P= 2S/c € Δp =ΔUc=uAcΔtc= uAΔt€ Δp = 2ΔUc= 2uAcΔtc= 2uAΔtChange of momentum in 100% absorption Change of momentum in 100% reflection € P =FA=ΔpΔtA= u =Sc10Example: Solar Energy The average intensity of the EM radiation from the Sun on Earth is S ~ 103 W/m2 What is the average radiation pressure for 100% absorption: What is the force exerted by EM radiation by the Sun on a surface of 1 m2 € P =Sc=103W m23⋅108ms= 3.3⋅10−6N m2€ F = PA = 3.3⋅10−6N m2⋅1m2= 3.3⋅10−6N11Antennas Antennas are essentially arrangement of conductors for transmitting and receiving radio waves. Parameters: gain, impedance, frequency, orientation, polarization, etc. half-wave antenna loop λ/4$λ/4$maximum strength microstrip 12Spectrum of EM Waves λf=c f (Hz) λ$VHF: 30-300 MHz UHF: 300 MHz -3.0 GHz Cell phone: 800/900/1800/1900 MHz Wifi: 2.4 GHz & 5 GHz Microwave Oven: 2.4 GHz Wireless phone: 2.4/5.8 GHz MHz GHz
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