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Physics 2112 Unit 19 Today’s Concepts: A) LC circuits and Oscillation Frequency B) Energy C) RLC circuits and Damping Electricity & Magnetism Lecture 19, Slide 1 LC Circuit - + I dI VL L dt L C Q VC C Q + - Q dI + L 0 Circuit Equation: C dt dQ I dt Solution to this DE: d 2Q Q 2 dt LC d 2Q 2 Q 2 dt where 1 LC Q(t ) Qmax cos(t + ) Electricity & Magnetism Lecture 19, Slide 2 Just like in 2111 mL d 2Q 2 Q dt2 2 d x 2 x 2 dt 1 LC L k k m C F = -kx a m x Same thing if we notice that 1 k C and mL Electricity & Magnetism Lecture 19, Slide 3 Also just like in 2111 1 2 1 2 mv kx 2 2 k F = -kx a m x Total Energy constant (when no friction present) 2 1Q 1 2 LI 2C 2 L C Total Energy constant (when no resistor present) Electricity & Magnetism Lecture 19, Slide 4 Q and I Time Dependence I L ++ C-- Charge and Current “90o out of phase” Electricity & Magnetism Lecture 19, Slide 5 CheckPoint 1 At time t = 0 the capacitor is fully charged with Qmax and the current through the circuit is 0. L C What is the potential difference across the inductor at t = 0? A) VL = 0 B) VL = Qmax/C C) VL = Qmax/2C since VL VC Electricity & Magnetism Lecture 19, Slide 6 CheckPoint 2 At time t = 0 the capacitor is fully charged with Qmax and the current through the circuit is 0. L C What is the potential difference across the inductor at when the current is maximum? A) VL = 0 B) VL = Qmax/C C) VL = Qmax/2C Electricity & Magnetism Lecture 19, Slide 7 CheckPoint 3 At time t = 0 the capacitor is fully charged with Qmax and the current through the circuit is 0. L C How much energy is stored in the capacitor when the current is a maximum ? A) U Qmax2/(2C) B) U Qmax2/(4C) C) U 0 Electricity & Magnetism Lecture 19, Slide 8 Example 19.1 (Charged LC circuit) After being left in position 1 for a long time, at t=0, the switch is flipped to V=12V position 2. L=0.82H All circuit elements are “ideal”. R=10W 1 2 C=0.01F What is the equation for the charge on the upper plate of the capacitor at any given time t? How long does it take the lower plate of the capacitor to fully discharge and recharge? Unit 19, Slide 9 Example 19.1 (Charged LC circuit) • What is the equation for the charge on the upper plate of the capacitor at any given time t? • How long does it take the lower plate of the capacitor to fully discharge and recharge? Conceptual Analysis Fill in all the terms in Q(t ) Qmax cos(t + ) Strategic Analysis Find initial current Use energy conservation to find Qmax Use L and C to find Use initial conditions to final Unit 19, Slide 10 Prelecture Question The switch is closed for a long time, resulting in a steady current Vb/R through the inductor. At time t = 0, the switch is opened, leaving a simple LC circuit. Which formula best describes the charge on the capacitor as a function of time? A. B. C. D. Q(t) = Qmaxcos(ωt) Q(t) = Qmaxcos(ωt + π/4) Q(t) = Qmaxcos(ωt + π/2) Q(t) = Qmaxcos(ωt + π) Electricity & Magnetism Lecture 19, Slide 11 CheckPoint 4 The capacitor is charged such that the top plate has a charge +Q0 and the bottom plate -Q0. At time t 0, the switch is closed and the circuit oscillates with frequency 500 radians/s. L ++ C - - L = 4 x 10-3 H = 500 rad/s What is the value of the capacitor C? A) C = 1 x 10-3 F B) C = 2 x 10-3 F C) C = 4 x 10-3 F Electricity & Magnetism Lecture 19, Slide 12 CheckPoint 5 closed at t = 0 L C +Q0 -Q0 Which plot best represents the energy in the inductor as a function of time starting just after the switch is closed? 1 2 U L LI 2 Electricity & Magnetism Lecture 19, Slide 13 CheckPoint 6 When the energy stored in the capacitor reaches its maximum again for the first time after t 0, how much charge is stored on the top plate of the capacitor? A) B) C) D) E) closed at t = 0 L C +Q0 -Q0 +Q0 +Q0 /2 0 -Q0/2 -Q0 Q is maximum when current goes to zero dQ dt Current goes to zero twice during one cycle I Electricity & Magnetism Lecture 19, Slide 14 Add Resistance 1 2 Switch is flipped to position 2. R d 2Q dQ 1 -L 2 -R - Q0 dt dt C Use “characteristic equation” at + bt + c 0 2 Unit 19, Slide 15 Damped Harmonic Motin Q QMax e - t cos( ' t + ) Damping factor R 2L Natural oscillation frequency o 1 LC Damped oscillation frequency '2 o2 - 2 Unit 19, Slide 16 Remember from 2111? Overdamped Critically damped Under damped Which one do you want for your shocks on your car? Mechanics Lecture 21, Slide 17 Example 19.2 (Charged LC circuit) 1 After being left in position 1 for a long time, at t=0, the switch is flipped to position 2. RL=7W V=12V The inductor now nonideal has some internal resistance. 2 L=0.82H C=0.01F R=10W What is the equation for the charge on the capacitor at any given time t? How long does it take the lower plate of the capacitor to fully discharge and recharge? Unit 19, Slide 18 V(t) across each elements The elements of a circuit are very simple: dI VL L dt VC V VL + VC + VR Q C dQ I dt VR IR This is all we need to know to solve for anything. But these all depend on each other and vary with time!! Electricity & Magnetism Lecture 19, Slide 19 How would we actually do this? Start with some initial V, I, Q, VL Now take a tiny time step dt (1 ms) VL dI dt L dQ Idt Q VC C Repeat… VR IR VL V - VR - VC What would this look like? Electricity & Magnetism Lecture 19, Slide 20 Example 19.3 In the circuit to the right • R1 = 100 Ω, • L1 = 300 mH, • L2 = 180 mH, • C = 120 μF and • V = 12 V. The positive terminal of the battery is indicated with a + sign. All elements are considered ideal. Q(t) is defined to be positive if V(a) – V(b) is positive. What is the charge on the bottom plate of the capacitor at time t = 2.82 ms? Conceptual Analysis Find the equation of Q as a function of time Strategic Analysis Determine initial current Determine oscillation frequency 0 Find maximum charge on capacitor Determine phase angle Electricity & Magnetism Lecture 19, Slide 21 Example 19.4 In the circuit to the right • R1 = 100 Ω, • L1 = 300 mH, • L2 = 180 mH, • C = 120 μF and • V = 12 V. The positive terminal of the battery is indicated with a + sign. All elements are considered ideal. Q(t) is defined to be positive if V(a) – V(b) is positive. What is the energy stored in the inductors at time t = 2.82 ms? Conceptual Analysis Use conservation of energy Strategic Analysis Find energy stored on capacitor using information from Ex 19.3 Subtract of the total energy calculated in the clicker question Electricity & Magnetism Lecture 19, Slide 22