Lecture 28Chapter 33EM Oscillations and ACReview• Can represent change in electric flux with a fictitious current called the displacement current, id• Ampere-Maxwell’s law becomesdtdiEdΦ=0εencEidtdsdB000µεµ+Φ=•∫rrencencdiisdB0,0µµ+=•∫rrReviewMaxwell’s 4 equations are• Gauss’ Law• Gauss’ Law for magnetism• Faraday’s Law• Ampere-Maxwell Law0εencqAdE =•∫rr0=•∫AdBrrdtdsdEBΦ−=•∫rrencEidtdsdB000µεµ+Φ=•∫rrEM Oscillations (1)• In this chapter study an RLC circuit• Review RC and RL circuits• Charge, current and potential difference grow and decay exponentially given by time constant, τCor τL• Look at LC circuitEM Oscillations (2)• RC circuit - resistor & capacitor in series– Charging up a capacitor – Discharging capacitor–whereRCC=τ)1(cteCqτ−−= Ecteqτ−=0qEM Oscillations (3)• RL Circuit – resistor & inductor in series– Rise of current – Decay of current–where()LteRτ−−= 1EiRLL=τLteiiτ−=0• Called Electromagnetic oscillations• Energy stored in E of capacitor and B of inductor EM Oscillations (4)• LC Circuit – inductor & capacitor in series• Find q, i and V vary sinusoidally with period T and angular frequency ω• E field of capacitor and Bfield of inductor oscillateCqUE22=22LiUB=EM Oscillations (5)EM Oscillations (6)• Cycle repeats at some frequency, fand thus angular frequency, ω• Ideal LC circuit, no R so oscillations continue indefinitely• Real LC, oscillations die away as energy goes into heat in Rfπω2=EM Oscillations (7)• Checkpoint #1 – A charged capacitor & inductor are connected in series at time t=0. In terms of period, T, how much later will the following reach their maximums:– q of capacitor T/2– VCwith original polarityT– Energy stored in E fieldT/2– The current T/4EM Oscillations (8)• LC circuits analogous to block-spring system • Total energy of block• Energy is conserved• Differentiating gives221221kxmvU +=0=dtdU()0221221=+=+=dtdxkxdtdvmvkxmvdtddtdUEM Oscillations (9)• Using • Substitute•Gives• Solution is 022=+ kxdtxdmdtdxv =022=+=+ kxvdtxdmvdtdxkxdtdvmv22dtxddtdv=)cos(φω+= tXx• X is the amplitude• ω is the angular frequency•φ is the phase constantmk=ωEM Oscillations (10)• Total energy of LC circuit• Total energy is constant• Differentiating givesCqLiUUUEB2222+=+=0=dtdU02222=+=+=dtdqCqdtdiLiCqLidtddtdUEM Oscillations (11)• Using • Substitute• Equation same form as block and spring• Solution isdtdqi =022=+=+ iCqdtqdLidtdqCqdtdiLi22dtqddtdi=0122=+ qCdtqdL• Q is the amplitude• ω is the angular frequency•φ is the phase constant)cos(φω+= tQqEM Oscillations (12)• Charge of LC circuit• Find current by • Amplitude I is dtdqi =)cos(φω+= tQq[])sin()cos(φωωφω+−=+= tQtQdtdiQIω=)sin(φω+−= tIiEM Oscillations (13)• What is ω for an LC circuit?• Substitute into • Find ω for LC circuit is)cos(222φωω+−= tQdtqd)cos(φω+= tQq0122=+ qCdtqdL0)cos(1)cos(2=+++−φωφωωtQCtQLLC1=ωEM Oscillations (14)• The phase constant, φφφφ, is determined by conditions at any certain time, t• If φφφφ= 0 at t = 0 thenq = Qi = 0)cos(φω+= tQq)sin(φω+−= tIiEM Oscillations (15)• The energy stored in an LC circuit at any time, t• Substitute •Using)sin(φω+−= tIi)cos(φω+= tQqEBUUU +=()φω+== tCQCqUE222cos22()φωω+== tQLLiUB2222sin22LC1=ω()φω+= tCQUB22sin2EM Oscillations (16)• For the case where φφφφ= 0• Maximum value for both • At any instant, sum is• When UE= max, UB= 0, and conversely, when UB= max, UE= 0 CQUUUEB22=+=()φω+= tCQUE22cos2()φω+= tCQUB22sin2CQUUBE22max,max,==EM Oscillations (17)• Checkpoint #2 – Capacitor in LC circuit has VC,max= 17V and UE,max= 160µJ. When capacitor has VC= 5V and UE= 10µJ , what are the a) emf across the inductor and b) the energy stored in the B field, UB?• Can apply the loop rule– Net potential difference around the circuit must be zeroA) vL = 5VB) UB = 160-10=150µJ)()( tvtvcL=)()(max,tUtUUBEE+=EM Oscillations (18)• Consider a RLC circuit – resistor, inductor and capacitor in series• Total electromagnetic energy, U = UE+ UB, is no longer constant • Energy decreases with time as it is transferred to thermal energy in the resistor • Oscillations in q, i and V are damped– Same as damped block and springEM Oscillations (19)• Resistor does not store electromagnetic energy so total energy at any time is • Rate of transfer to thermal energy is (minus sign means U is decreasing)• Differentiating givesCqLiUUUEB2222+=+=RidtdU2−=RidtdqCqdtdiLidtdU2−=+=EM Oscillations (20)• Use relations• Differential equation for damped RLC circuit is• Solution dtdqi =RidtdqCqdtdiLidtdU2−=+=22dtqddtdi=0122=++ qCdtdqRdtqdL)cos(2/φω+′=−tQeqLRtEM Oscillations (21)• Where • Charge in RLC circuit is sinusoidal but with an exponentially decaying amplitude• Damped angular frequency, ω´, is always less than ω of the undamped oscillations22)2/( LR−=′ωω)cos(2/φω+′=−tQeqLRtLC1=ωLRtQe2/−EM Oscillations (22)• Find UEas function of time• Total energy decreases as
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