Physics 2102 Jonathan Dowling Lecture 29 WED 25 MAR 09 Ch 31 1 4 Electrical Oscillations LC Circuits Alternating Current EXAM 03 6PM THU 02 APR 2009 The exam will cover Ch 28 second half through Ch 32 1 3 displacement current and Maxwell s equations The exam will be based on HW08 HW11 Final Day to Drop Course FRI 27 MAR What are we going to learn A 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 in Physics Oscillators are very useful in practical applications for instance to keep time or to focus energy in a system All oscillators can store energy in more than one way and exchange it back and forth between the different storage possibilities For instance in pendulums and swings one exchanges energy between kinetic and potential form We have studied that inductors and capacitors are devices that can store electromagnetic energy energy In the inductor it is stored in a B field in the capacitor in an E field PHYS2101 A Mechanical Oscillator U tot U kin U pot const U tot 1 1 2 2 mv k x 2 2 dU tot 1 dv 1 dx 0 m 2v k 2x dt 2 dt 2 dt dv m kx 0 dt Solution v x t a v t x t Newton s law d 2x m 2 k x 0 F ma dt k x t x0 cos t 0 m x0 amplitude 0 frequency phase PHYS2101 An Electromagnetic LC Oscillator Capacitor initially charged Initially current is zero energy is all stored in the capacitor Energy Conservation U tot U B U E A current gets going energy gets split between the capacitor and the inductor 2 1 2 1q U B L i U E 2 2C Capacitor discharges completely yet current keeps going Energy is all in the inductor The magnetic field on the coil starts to collapse which will start to recharge the capacitor U tot 1 2 1 q2 Li 2 2C Finally we reach the same state we started with with opposite polarity and the cycle restarts Electric Oscillators the Math U tot U B U E U tot 1 2 1 q2 Li 2 2C dU tot 1 di 1 dq 0 L 2i 2q dt 2 dt dt 2C Energy Cons di 1 VL VC 0 L q dt C Or loop rule Both give Diffy Q Solution to Diffy Q d 2q q 0 L 2 dt C q q0 cos t 0 1 LC i q t i t q t LC Frequency In Radians Sec i q t q0 sin t 0 i t q t 2 q0 cos t 0 Electric Oscillators the Math q q0 cos t 0 i q t q0 sin t 0 i t q t 2 q0 cos t 0 Energy as Function of Time 1 1 2 2 U B L i L q0 cos t 0 2 2 1 q 1 2 UE q cos t 0 0 2 C 2C 2 Voltage as Function of Time VL Li t q0 sin t 0 2 2 1 1 VC q t q0 cos t 0 C C Analogy Between Electrical And Mechanical Oscillations d 2q q 0 L 2 dt C 1 LC d 2x m 2 k x 0 dt k m q q0 cos t 0 x t x0 cos t 0 i q t q0 sin t 0 v x t x0 sin t 0 i t q t q0 cos t 0 a x t 2 x0 cos t 0 2 q x i v 1 C k L m Charqe q Position x Current i q Velocity v x D Current i q Acceleration a v x LC Circuit Conservation of Energy q q0 cos t 0 1 5 1 0 5 0 Time 0 5 Charge Current 1 dq i q0 sin t 0 dt 1 2 1 U B Li L 2 q02 sin 2 t 0 2 2 1 q2 1 2 UE q0 cos 2 t 0 2C 2C 1 5 1 2 And remembering that 1 0 8 0 6 0 4 0 2 0 Time Energy in capacitor Energy in coil 1 cos x sin x 1 and LC 2 U tot 2 1 2 UB UE q0 2C The energy is constant and equal to what we started with Example 1 Tuning a Radio Receiver The inductor and capacitor in my car radio are usually set at L 1 mH C 3 18 pF Which is my favorite FM station a KLSU 91 1 b WRKF 89 3 c Eagle 98 1 WDGL FM radio stations frequency is in MHz 1 LC 1 6 1 10 3 18 10 5 61 10 8 rad s f 2 8 93 10 7 Hz 89 3 MHz 12 rad s Example 2 In an LC circuit L 40 mH C 4 F At t 0 the current is a maximum When will the capacitor be fully charged for the first time 1 5 1 0 5 0 Time 0 5 1 1 5 Charge Current 1 1 rad s LC 16x10 8 2500 rad s T period of one complete cycle T 2 2 5 ms Capacitor will be charged after T 1 4 cycle i e at t T 4 0 6 ms Example 3 In the circuit shown the switch is in position a for a long time It is then thrown to position b Calculate the amplitude q0 of the resulting oscillating current 1 mH 1 F b E 10 V a i q0 sin t 0 Switch in position a q CV 1 F 10 V 10 C Switch in position b maximum charge on C q0 10 C So amplitude of oscillating current 1 q0 10 C 0 316 A 1mH 1 F Example 4 In an LC circuit the maximum current is 1 0 A If L 1mH C 10 F what is the maximum charge q0 on the capacitor during a cycle of oscillation q q0 cos t 0 dq i q0 sin t 0 dt Maximum current is i0 q0 Maximum charge q0 i0 Angular frequency 1 LC 1mH 10 F 1 2 10 8 1 2 104 rad s Maximum charge is q0 i0 1A 104 rad s 10 4 C
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