Lecture 28 Chapter 33 EM Oscillations and AC Review r r d E B ds 0 0 dt 0ienc Can represent change in electric flux with a fictitious current called the displacement current id d id 0 Ampere Maxwell s law becomes r r B ds 0id enc 0ienc dt E Review Maxwell s 4 equations are Gauss Law r q enc r E dA 0 Gauss Law for magnetism r r B dA 0 r r d B Faraday s Law E ds dt r r d E Ampere Maxwell Law B ds 0 0 0ienc dt EM 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 C or L Look at LC circuit EM Oscillations 2 RC circuit resistor capacitor in series Charging up a capacitor q CE 1 e t c Discharging capacitor q q0 e where t c C RC EM Oscillations 3 RL Circuit resistor inductor in series Rise of current E t L i 1 e R Decay of current i i0e where L t L L R EM Oscillations 4 LC Circuit inductor capacitor in series Find q i and V vary sinusoidally with period T and angular frequency E field of capacitor and B field of inductor oscillate Called Electromagnetic oscillations Energy stored in E of capacitor and B of inductor 2 2 q UE 2C U B Li 2 EM Oscillations 5 EM Oscillations 6 Cycle repeats at some frequency f and thus angular frequency 2 f Ideal LC circuit no R so oscillations continue indefinitely Real LC oscillations die away as energy goes into heat in R 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 VC with original polarity T Energy stored in E field T 2 The current T 4 EM Oscillations 8 LC circuits analogous to block spring system Total energy of block U mv kx 1 2 2 1 2 2 Energy is conserved dU Differentiating gives dt 0 dx dU d 1 2 1 2 dv 2 mv 2 kx mv kx 0 dt dt dt dt EM Oscillations 9 dx v dt Using Substitute Gives dv d 2x dt dt 2 2 d x dx dv kx mv 2 kxv 0 mv dt dt dt 2 d x m kx 0 2 dt Solution is x X cos t X is the amplitude is the angular frequency k is the m phase constant EM Oscillations 10 Total energy of LC circuit Li 2 q 2 U UB UE 2 2C Total energy is constant dU 0 dt Differentiating gives dU d Li q di q dq Li 0 dt dt 2 2C dt C dt 2 2 EM Oscillations 11 Using dq i dt Substitute di d 2 q 2 dt dt 2 d q q di q dq Li Li 2 i 0 dt C dt dt C Equation same form as block and spring 2 Solution is d q 1 L 2 q 0 dt C q Q cos t Q is the amplitude is the angular frequency is the phase constant EM Oscillations 12 Charge of LC circuit q Q cos t Find current by dq i dt d i Q cos t Q sin t dt Amplitude I is I Q i I sin t EM Oscillations 13 What is for an LC circuit q Q cos t Substitute into d 2q 2 Q cos t 2 dt 2 d q 1 L 2 q 0 dt C 1 L Q cos t Q cos t 0 C 2 Find for LC circuit is 1 LC EM Oscillations 14 The phase constant is determined by conditions at any certain time t q Q cos t i I sin t If 0 at t 0 then q Q i 0 EM Oscillations 15 The energy stored in an LC circuit at any time t Substitute U U Using B E U UB UE q Q cos t i I sin t q2 Q2 cos 2C 2C 2 t Li 2 L 2 2 Q sin 2 2 1 LC U B 2 t Q2 sin 2C 2 t EM Oscillations 16 For the case where 0 U U E B Q2 cos 2C Q2 sin 2C 2 t 2 t Maximum value for both U E max U B max Q 2C At any instant sum is U U B U E Q2 2C When UE max UB 0 and conversely when UB max UE 0 2 EM Oscillations 17 Checkpoint 2 Capacitor in LC circuit has VC max 17V and UE max 160 J When capacitor has VC 5V and UE 10 J 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 zero vL t vc t A vL 5V UE max UE t UB t B UB 160 10 150 J 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 spring EM Oscillations 19 Resistor does not store electromagnetic energy so total energy at any time is 2 2 Li q U UB UE 2 2C Rate of transfer to thermal dU 2 i R energy is minus sign means dt U is decreasing Differentiating gives di q dq dU 2 Li i R dt dt C dt EM Oscillations 20 di q dq dU 2 Li i R dt C dt dt dq Use relations i dt 2 di d q 2 dt dt Differential equation for damped RLC circuit is 2 dq 1 d q L R q 0 2 dt dt C Solution q Qe Rt 2 L cos t EM Oscillations 21 Rt 2 L q Qe cos t Where R 2L 2 2 1 LC Charge in RLC circuit is sinusoidal but with an exponentially decaying amplitude Rt 2 L Qe Damped angular frequency is always less than of the undamped oscillations EM Oscillations 22 q Qe Rt 2 L cos t Find UE as function of time 2 2 q Q Rt L UE e cos2 t 2C 2C 2 Total energy decreases as U tot Q Rt L e 2C
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