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Todays Concepts: A) LC circuits and Oscillation Frequency B) Energy C) RLC circuits and Damping Physics 2112 Unit 19 Electricity & Magnetism Lecture 19, Slide 1

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C I L Q LC Circuit Electricity & Magnetism Lecture 19, Slide 2 Circuit Equation: where Solution to this DE:

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C L k x m F F = -kx a Same thing if we notice that and Electricity & Magnetism Lecture 19, Slide 3 Just like in 2111

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k x m F F = -kx a Electricity & Magnetism Lecture 19, Slide 4 Also just like in 2111 Total Energy constant (when no friction present) C L Total Energy constant (when no resistor present)

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C L I Q and I Time Dependence Electricity & Magnetism Lecture 19, Slide 5 Charge and Current 90 o out of phase

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What is the potential difference across the inductor at t = 0 ? A) V L = 0 B) V L = Q max /C C) V L = Q max /2C At time t = 0 the capacitor is fully charged with Q max and the current through the circuit is 0. L C since V L V C CheckPoint 1 Electricity & Magnetism Lecture 19, Slide 6 Inductor does not need current to have voltage drop! Its not like a resistor!

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CheckPoint 2 Electricity & Magnetism Lecture 19, Slide 7 At time t = 0 the capacitor is fully charged with Q max 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) V L = 0 B) V L = Q max /C C) V L = Q max /2C Q is max but dQ/dt is 0!

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How much energy is stored in the capacitor when the current is a maximum ? A) U Q max 2 /(2C) B) U Q max 2 /(4C) C) U 0 CheckPoint 3 Electricity & Magnetism Lecture 19, Slide 8 At time t = 0 the capacitor is fully charged with Q max and the current through the circuit is 0. L C Max current means all of U is in inductor!

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L=0.82H Example 19.1 (Charged LC circuit) Unit 19, Slide 9 1 2 After being left in position 1 for a long time, at t=0, the switch is flipped to position 2. All circuit elements are ideal. C=0.01F V=12V 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? R=10

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Example 19.1 (Charged LC circuit) Unit 19, Slide 10 How long does it take the lower plate of the capacitor to fully discharge and recharge? Conceptual Analysis Fill in all the terms in Strategic Analysis Find initial current Use energy conservation to find Q max Use L and C to find Use initial conditions to final What is the equation for the charge on the upper plate of the capacitor at any given time t?

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The switch is closed for a long time, resulting in a steady current V b /R through the inductor. A.Q(t) = Q max cos(ωt) B.Q(t) = Q max cos(ωt + π/4) C.Q(t) = Q max cos(ωt + π/2) D.Q(t) = Q max cos(ωt + π) Prelecture Question Electricity & Magnetism Lecture 19, Slide 11 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?

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., The capacitor is charged such that the top plate has a charge Q 0 and the bottom plate Q 0. At time t 0, the switch is closed and the circuit oscillates with frequency 500 radians/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 L C L = 4 x 10 -3 H = 500 rad/s CheckPoint 4 Electricity & Magnetism Lecture 19, Slide 12

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Which plot best represents the energy in the inductor as a function of time starting just after the switch is closed? L C closed at t = 0 Q 0 CheckPoint 5 Electricity & Magnetism Lecture 19, Slide 13

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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) Q 0 B) Q 0 /2 C) 0 D) Q 0 /2 E) Q 0 CheckPoint 6 Electricity & Magnetism Lecture 19, Slide 14 L C closed at t = 0 Q 0 Current goes to zero twice during one cycle Q is maximum when current goes to zero

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Add Resistance Unit 19, Slide 15 1 2 Switch is flipped to position 2. R Use characteristic equation

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Damped Harmonic Motin Unit 19, Slide 16 Damping factor Damped oscillation frequency Natural oscillation frequency

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Remember from 2111? Mechanics Lecture 21, Slide 17 Overdamped Critically damped Under damped Which one do you want for your shocks on your car?

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L=0.82H Example 19.2 (Charged LC circuit) Unit 19, Slide 18 1 2 After being left in position 1 for a long time, at t=0, the switch is flipped to position 2. The inductor now non- ideal has some internal resistance. What is the equation for the charge on the capacitor at any given time t? C=0.01F V=12V R L =7 R=10 How long does it take the lower plate of the capacitor to fully discharge and recharge ?

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The elements of a circuit are very simple: 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 V(t) across each elements

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Start with some initial V, I, Q, V L Now take a tiny time step dt (1 ms) Repeat… How would we actually do this? Electricity & Magnetism Lecture 19, Slide 20 What would this look like?

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In the circuit to the right R 1 = 100 Ω, L 1 = 300 mH, L 2 = 180 mH, C = 120 μF and V = 12 V. Example 19.3 Electricity & Magnetism Lecture 19, Slide 21 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 Find maximum charge on capacitor Determine phase angle

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In the circuit to the right R 1 = 100 Ω, L 1 = 300 mH, L 2 = 180 mH, C = 120 μF and V = 12 V. Example 19.4 Electricity & Magnetism Lecture 19, Slide 22 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

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