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Lec20 - Lecture 20-1 d Q 1 Q = 0 0 = dt 2 LC 1 LC 2 LC...

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Lecture 20 Lecture 20 - 1 LC Oscillations 2 2 1 , , 2 2 E B Q dQ U U LI I C dt = = = No Resistance = No dissipation 2 0 2 0 0 2 1 1 0 d Q Q dt LC LC f ω ω π + = = =
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Lecture 20 Lecture 20 - 2 Mechanical Analogy 2 2 1 1 , , 2 2 dx U kx K mv v dt = = = max 0, U K K = = max , 0 U U K = = max , 0 U U K = = . E const U K = = + harmonic oscillator with 0 k m ω = No friction = No dissipation 2 0 2 0 d x k k x dt m m ω + = = 0 0 / 2 f ω π =
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Lecture 20 Lecture 20 - 3 More on LC Oscillations Energy stored in capacitor: 2 2 0 1 ( ) cos 2 E peak U t Q t C ω = t 0 E U 0 t B U Energy stored in inductor: 2 2 2 0 0 1 ( ) sin 2 B peak U t L Q t ω ω = 0 1 LC ω = where 2 2 0 1 ( ) sin 2 B peak U t Q t C ω = 2 ( ) ( ) 2 peak E B Q U t U t C + = so 0 0 sin peak dQ I Q t dt ω ω = = − Charge and current: 0 cos peak Q Q t ω = (with δ =0) Period is half that of Q(t)
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Lecture 20 Lecture 20 - 4 Series RLC Circuits The resistance R may be a separate component in the circuit, or the resistance inherent in the inductor (or other parts of the circuit) may be represented by R . Finite R Energy dissipation damped oscillation only if R is “small” 0 dI Q L IR dt C + + = 2 2 1 0 d Q dQ L R Q dt dt C + + = 2 2 0 d x dx m b kx dt dt + + = multiply by I 2 2 2 1 0 2 2 d d Q LI I R dt dt C + + = For large R
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Lecture 20 Lecture 20 -
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