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# MSU PHY 184 - Lecture31_white

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Lecture 31Chapter 34Electromagnetic WavesReview• For an RLC circuit – Voltages add up to emf– Maximum current given by– Impedance defined as– Phase constant defined as LCRvvv ++=EZImE=22)(CLXXRZ −+=LXdLω=CXdCω1=RXXCL−=φtanReview• For RLC circuit, resonance and the max current I occurs when ωd= ω• For an ac circuit, define rmsvalues • Average power dissipated to thermal energy LCd1==ωω2IIrms=RIPrmsavg2=φcosrmsrmsavgIP E=2VVrms=2EE =rmsReview• Transformer –– 2 coils (primary and secondary) wound around same iron core– Transformation voltage and current are related to ratio of the number of turns in the coils – Equivalent resistance seen by generatorPSPSNNVV =RNNRSPeq2=SPPSNNII =EM Waves (1) • Electromagnetic waves –– Beam of light is a traveling wave of E and B fields – All waves travel through free space with same speedEM Waves (2) • Generate electromagnetic (EM) waves –– Sinusoidal current in RLC causes charge and current to oscillate along rods of antenna with angular frequency ω– Changing E and B fields form EM wave that travels away from antenna at speed of light, cLC1=ωEM Waves (3) • E and B fields change with time and have features:– E and B fields ⊥⊥⊥⊥ to direction of wave’s travel – transverse wave– E field is ⊥⊥⊥⊥ B field– Direction of wave’s travel is given by cross product– E and B fields vary• Sinusodially• With same frequency and in phaseBErr×EM Waves (4) • Write E and B fields as sinusoidal functions of position x (along path of wave) and time t• Angular frequency ω and angular wave number k• E and B components cannot exist independently)sin( tkxEEmω−=)sin( tkxBBmω−=fπω2=λπ2=kEM Waves (5) • Speed of wave is• Using definition of ωand k, velocity is • In vacuum EM waves move at speed of lightfπω2=λπ2=kkvω=smc /1038×=λλππωffkv ===/22λfcv ==EM Waves (6) • Use Faraday’s and Maxwell’s laws of induction• Can prove that speed of light c is given by (proof done in section 34-3)• Light travels at same speed regardless of what reference frame its measured in001εµ=cmmBEc =smc /1038×=dtdsdEBΦ−=•∫rrdtdsdBEΦ=•∫00εµrrEM Waves (7) • EM waves can transport energy and deliver it to an object it falls on• Rate of energy transported per unit area is given by Poynting vector, S, and defined as• SI unit is W/m2• Direction of S gives wave’s direction of travel BESrrr×=01µEM Waves (8) • Magnitude of S is given by• Found relation• Rewrite S in terms of E since most instruments measure Ecomponent rather than B• Instantaneous energy flow rate is mmBEc =cEES01µ=EBS01µ=201EcSµ=EM Waves (9) • Usually want time-averaged value of S also called intensity I• Average value over full cycle of • Use the rms value• Rewrite average S or intensity as2mrmsEE =aveaveavgareapowerareatimeenergySI===/2/1sin2=θ[] []avgmavgtkxEcEcI )(sin1122020ωµµ−==201rmsEcIµ=EM Waves (10) • Find intensity, I , of point source which emits light isotropically –equal in all directions• Find I at distance r from source• Imagine sphere of radius r and area• I decreases with square of distance 24 rAπ=24 rPAreaPowerISπ==aveaveavgareapowerareatimeenergySI===/EM Waves (11) • Checkpoint #2 – Have an E field shown in picture. A wave is transporting energy in the negative z direction. What is the direction of the B field of the wave?• Poynting vector gives• Use right-hand rule to find B fieldPositive x

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