1EM WavesThis LectureMore on EM wavesEM spectrumPolarizationFrom previous LectureDisplacement currentsMaxwell’s equationsEM Waves2Reminders on wavesTraveling waves on a string along x obey the wave equation: y=wave functionGeneral solution : y(x,t) = f1(x-vt) + f2(x+vt) y = displacementTraveling wave: superposition of sinusoidal waves (produced by a source that oscillates with simple harmonic motion): y(x,t) = A sin(kx-ωt) y(x,t) = sin(kx+ ωt) A = amplitudek = 2π/λ = wave number λ = wavelength f = frequency T = 1/f = periodω = 2πf=2π/T angular frequencyλ€ ∂2y(x,t)∂x2=1v2∂2y(x,t)∂t2pulse traveling along +x pulse traveling along -xvMaxwell equations in vacuumin the absence of charges (q=0) and conduction currents (I=0)€ E ⋅ dA = 0S∫ (Gauss' Law) B ⋅ dA = 0S∫ E ⋅ ds = −dΦBdt (Faraday - Henry) B ⋅ dsL∫= L∫µ0ε0dΦEdt (Ampere - Maxwell law)From these equations we get EM wave equations traveling in vacuum!4λEM waves from Maxwell equationsE x B direction of cSolutions of these equations:sinusoidal traveling transverse waves propagating along x E = Emax cos (kx – ωt) B = Bmax cos (kx – ωt)•E and B are perpendicular oscillating vectorsE · B = 05E and B are orthogonalAn easy way to understand this:ΦB = B A cosθ1. Max B flux θ=0 => also circular E is largest!2. Less flux3. Null flux θ=90° => circular E smallest!B parallel to area normal and E perpendicular to circuit so E ⊥ B € E ⋅ ds = -dΦBdt∫E orthogonal to B!123θIIncreasing B-fieldEE · B = 0Shown below is the E-field of an EM wave broadcast at 30 MHz and traveling to the right.What is the direction of the magnetic field during the first λ/2? What is the wave length?Quick Quiz on EM Waves Ex1) Into the page 2) out of the page1) 10 m 2) 5 m λ/2xzyEBloop in xy planeloop in xz planeloop in yz planeA B CWhich orientation will have the largest induced emf? Quick Quiz pn EM wavesImportant Relation between E and BE = Emax cos (kx – ωt) B = Bmax cos (kx – ωt) First derivatives: € ∂E∂x= −kEmaxsin(kx −ωt)∂B∂t=ωBmaxsin(kx −ωt)∂E∂x= −∂B∂tThis relation comes from Maxwell’s equations!From:EM Waves generators: Antennas2 rods connected to alternate current generator; charges oscillate between the rods (a)As oscillations continue, the rods become less charged, the field near the charges decreases and the field produced at t = 0 moves away from the rod (b)The charges and field reverse (c)The oscillations continue (d)Sources of EM waves: oscillating charges, accelerated/decellerated charges, electron transitions between energy levels in atoms, nuclei and moleculesThe EM SpectrumX-rays: ~10-12 -10-8 msource: deceleration of high-energy electrons striking a metal targetDiagnostic tool in medicineSource: atoms and molecules Human eyeVisible range from red (700 nm) to violet (400 nm)Radio: λ ~ 10 - 0.1 mSources: charges accelerating through conducting wires Radio and TVMicrowaves: λ ~10-4 -0.3 m sources: electronic devicesradar systems, MW ovensInfrared: λ ~ 7 x 10-7-10-3 mSources: hot objects and moleculesUV λ~ 6 x 10-10 - 4 x 10-7 mMost UV light from the sun is absorbed in the stratosphere by ozoneGamma rays: λ~ 10-14- 10-10 mSource: radioactive nuclei, cause serious damage to living tissuesPoynting vectorRate at which energy flows through a unit area perpendicular to direction of wave propagation This is the power per unit area (J/s.m2 = W/m2)Its direction is the direction of propagation of the EM waveMagnitude: Magnitude is time dependentreaches a max at the same instant as E and B do€ S =EBµ0=E2cµ0E/B=cEnergy density of E and B field In a parallel plate capacitor: Similarly for a solenoid with current: € C =ε0Ad€ U =12CV2=12ε0AdE2d2⇒ uE=UAd=12ε0E2True for any geometryTrue for any geometry€ uB=B22µ0Energy carried by EM waves€ uE=12ε0E2= uB=B22µ0=E22c2µ0Total instantaneous energy density of EM waves u =uE + uB = 1/2 εoE2 + B2 /(2µo)Since B = E/c andEM waves carry energy!In a given volume, the energy is shared equally by the two fields€ uE= uBThe average energy density over one or more cycles of oscillations is:€ uav= 2uE ,av=ε0E2=12ε0Emax2since <sin2(kx - ωt)> = 1/2E x BIntensity and Poynting vectorLet’s consider a cylinder with axis along x of area A and length L and the time for the wave to travel L is Δt=L/cThe average power of the EM wave in the cylinder is:The intensity is € Pav=UavΔt=uavALΔt= uavAcE/B=c€ I =PavA=12ε0cEmax2=EmaxBmax2µ0average power per unit area (units W/m2) € I= SavI ∝ E2Radiation pressure and momentumComplete absorption on a surface: total transferred momentum p = U / c and prad=Sav/cRadiation momentum: Radiation Pressure Perfectly reflecting surface: p = 2U/c and prad = 2Sav/c€ F = ma = mdvdt=dpdt€ p = force × time = force × distance ×timedistan ce=energy velocity=Ucthe radiation momentum is the radiation energy/velocity€ prad=FA=pAΔt=UcAΔt=PowercA=Ic€ Sav= I =PowerARadiation pressure from the SunSolar intensity at the Earth (~1.5 x 108 km far from the Sun): ~1350 W/m2Direct sunlight pressure I/c ~4.5 x 10-6 N/m2If the sail is a reflecting mirror prad = 2 x 4.5 x 10-6 N/m2Can be used by spacecrafts for propulsion as wind for sailing boats!What is the sail area needed to accelerate a 104 kg spacecraft at a = 0.01 m/s2 assuming perpendicular incidence of the radiation on the sail?A = F/prad = ma/prad = 107 m2€ prad=FA=Power / Ac=Ic€ =SavcPolarization of Light Waves (34.8)Linearly polarized waves: E-field oscillates at all times in the plane of polarizationLinearly polarized light: E-field has one spatial orientationUnpolarized light: E-field in random directions. Superpositionof waves with E vibrating in many different directionsCircular and elliptical polarizationCircularly polarized light: superposition of 2 waves of equal amplitude with orthogonal linear polarizations, and 90˚ out of phase. The tip of E describes a circle (counterclockwise = RH and clockwise=LH depending on y component ahead or behind)The electric field rotates in time with constant magnitude.If amplitudes differ ⇒ elliptical polarizationProducing polarized lightPolarization by selective absorption:
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