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LSU PHYS 2102 - Electrical Oscillations

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Physics 2102Jonathan DowlingLecture 29: WED 25 MAR 09Lecture 29: WED 25 MAR 09Ch. 31.1Ch. 31.1––4: Electrical Oscillations, LC4: Electrical Oscillations, LCCircuits, Alternating CurrentCircuits, Alternating CurrentEXAM 03: 6PM THU 02 APR 2009The exam will cover:Ch.28 (second half) throughCh.32.1-3 (displacement current,and Maxwell's equations).The exam will be based on:HW08 – HW11Final Day to Drop Course: FRI 27 MARWhat are we going to learn?What are we going to learn?A road mapA road map• Electric charge Electric force on other electric charges Electric field, and electric potential• Moving electric charges : current• Electronic circuit components: batteries, resistors, capacitors• Electric currents  Magnetic field Magnetic force on moving charges• Time-varying magnetic field  Electric Field• More circuit components: inductors.• Electromagnetic waves  light waves• Geometrical Optics (light rays).• Physical optics (light waves)Oscillators are very useful in practicalapplications, for instance, to keep time, orto focus energy in a system.All oscillators can store energy inmore than one way and exchangeit back and forth between thedifferent storage possibilities. Forinstance, in pendulums (and swings)one exchanges energy betweenkinetic and potential form.Oscillators in PhysicsOscillators in PhysicsWe have studied that inductors and capacitors are devicesthat can store electromagnetic energyelectromagnetic energy. In the inductor it isstored in a B field, in the capacitor in an E field.Utot= Ukin+ Upot= constUtot=12m v2+12k x2dUtotdt= 0 =12m 2vdvdt!"#$%&+12k 2xdxdt!"#$%&v =!x (t )a =!v (t ) =!!x (t )! mdvdt+ k x = 0)cos()( :Solution00!"+= txtxphase : frequency : amplitude :00!"xmk=!PHYS2101: A Mechanical OscillatorPHYS2101: A Mechanical Oscillator022=+ xkdtxdmNewton’s lawF=ma!The magnetic field on the coil starts to collapse,which will start to recharge the capacitor.Finally, we reach the same state we started with (withopposite polarity) and the cycle restarts.PHYS2101 An Electromagnetic LC OscillatorPHYS2101 An Electromagnetic LC OscillatorCapacitor discharges completely, yet current keeps going.Energy is all in the inductor.Capacitor initially charged. Initially, current is zero,energy is all stored in the capacitor.A current gets going, energy gets split between thecapacitor and the inductor.Energy!Conservation:!Utot= UB+ UEUtot=12L i2+12qC2UB=12L i2!!!!!UE=12qC2Utot= UB+ UEUtot=12L i2+12qC2dUtotdt= 0 =12L 2ididt!"#$%&+12C2qdqdt!"#$%&VL+ VC= 0 = Ldidt!"#$%&+1Cq( )i =!q (t)!i (t) =!!q (t)CqdtqdL +=220!"1LCq = q0cos(!t +"0)Electric Oscillators: the MathElectric Oscillators: the MathOr loop rule!i =!q (t) = "q0#sin(#t +$0)!i (t) =!!q (t) = "#2q0cos(#t +$0)Energy Cons.Both give Diffy-Q:Solution to Diffy-Q:LC FrequencyIn Radians/SecUB=12L i[ ]2=12L q0!cos(!t +"0)[ ]2VL= L!i (t) ="2q0sin("t +#0)$%&'2q = q0cos(!t +"0)Electric Oscillators: the MathElectric Oscillators: the Mathi =!q (t) = "q0#sin(#t +$0)!i (t) =!!q (t) = "#2q0cos(#t +$0)UE=12q[ ]C2=12Cq0cos(!t +"0)[ ]2Energy as Function of TimeVoltage as Function of TimeVC=1Cq(t)[ ]=1Cq0cos(!t +"0)[ ]022=+ xkdtxdmAnalogy Between Electrical And Mechanical Oscillationsq ! x 1 / C ! ki ! v L ! mLC1=!)cos()(00!"+= txtxmk=!CqdtqdL +=220q = q0cos(!t +"0)i =!q (t) = "q0#sin(#t +$0)!i (t) =!!q (t) = "#2q0cos(#t +$0)v =!x (t ) = "x0#sin(#t +$0)a =!!x (t ) = "#2x0cos(#t +$0)Charqe q -> Position xCurrent i=q’ -> Velocity v=x’D-Current i’=q’’-> Acceleration a=v’=x’’-1.5-1-0.500.511.5TimeChargeCurrent)cos(00!"+= tqq)sin(00!""+#== tqdtdqiUB=12Li2=12L!2q02sin2(!t +"0)00.20.40.60.811.2TimeEnergy in capacitorEnergy in coilUE=12qC2=12Cq02cos2(!t +"0)LCxx1 and ,1sincos that,grememberin And22==+!Utot= UB+ UE=12Cq02The energy is constant and equal to what we started with.LC Circuit: Conservation of EnergyLC Circuit: Conservation of EnergyExample 1 : Tuning a Radio ReceiverExample 1 : Tuning a Radio Receiver The inductor and capacitorin my car radio are usuallyset at L = 1 mH & C = 3.18pF. Which is my favorite FMstation?(a) KLSU 91.1(b) WRKF 89.3(c) Eagle 98.1 WDGLFM radio stations: frequency is in MHz.!=1LC=11 " 10#6" 3.18 " 10#12rad/s= 5.61 " 108rad/sf =!2"= 8.93 # 107Hz= 89.3!MHzExampleExample 22• In an LC circuit,L = 40 mH; C = 4 µF• At t = 0, the current isa maximum;• When will the capacitorbe fully charged for thefirst time?!=1LC=116x10"8rad/s• ω = 2500 rad/s• T = period of onecomplete cycle•T = 2π/ω = 2.5 ms• Capacitor will becharged after T=1/4cycle i.e at• t = T/4 = 0.6 ms-1.5-1-0.500.511.5TimeChargeCurrentExample 3Example 3• In the circuit shown, theswitch is in position “a” for along time. It is then thrownto position “b.”• Calculate the amplitude ωq0of the resulting oscillatingcurrent.• Switch in position “a”: q=CV = (1 µF)(10 V) = 10 µC• Switch in position “b”: maximum charge on C = q0 = 10 µC• So, amplitude of oscillating current =!q0=1(1mH)(1µF)(10µC) =0.316 A)sin(00!""+#= tqibaE=10 V1 mH1 µFExample 4Example 4In an LC circuit, the maximum current is 1.0 A.If L = 1mH, C = 10 µF what is the maximum charge q0 onthe capacitor during a cycle of oscillation?)cos(00!"+= tqq)sin(00!""+#== tqdtdqiMaximum current is i0=ωq0 → Maximum charge: q0=i0/ωAngular frequency ω=1/√LC=(1mH 10 µF)–1/2 = (10-8)–1/2 = 104 rad/sMaximum charge is q0=i0/ω = 1A/104 rad/s = 10–4


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