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p240_ct22_f07

# p240_ct22_f07 - Physics 240 Fall 2007 Lecture#22 No...

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Physics 240 Fall 2007 Lecture #22 Dr. Dave Winn 2405 Randall Lab [email protected] No discussion tomorrow (11/21)

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Essential ideas from last time Electrical oscillations in LC circuits Frequency of oscillation ω = 1/ LC Total energy is constant Energy is traded between E (in capacitor) and B (in inductor) fields Charge on C and current in L both oscillate: Q(t) = Q 0 cos( ω t+ ϕ ) I(t) = - ω Q 0 sin( ω t+ ϕ ) These oscillations are “out of phase”: when charge is large, current is small, when charge is small, current is large.
Real oscillators Real oscillators don’t run forever. They gradually lose energy In mechanical oscillators, friction drains energy In electrical oscillators, resistance drains energy Time it takes things to stop depends on what keeps it going (inertia and L) compared to what stops it (friction and R). Timescale for mechanical damping: ~inertia / friction Timescale for electrical damping: inductance / res. (~L/R) Just like a normal RL circuit!

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The Mechanical Example Mass on a spring is the oscillator, friction provides the damping If damping is weak, many oscillations before dying away, if strong, no oscillations will occur Oscillation frequency is shifted, it oscillates more slowly The equations: ( ) ( ) ( ) 2 2 2 2 0 2 2 2 2 ) cos( 0 m b m k m b t e X t x kx dt dx b dt x d m ma bv kx ma F m bt = = + = = + + = = ω ω ω φ ω
The Electrical Example Oscillator is the LC circuit, damping from resistance If damping is weak, many oscillations before dying away, if strong, no oscillations will occur Oscillation frequency is shifted lower The equations: ( ) ( ) ( ) 2 2 2 2 2 2 2 1 2 ) cos( 0 0 L R LC L R t Qe t q C q dt dq R dt q d L dt di L iR C q L Rt = = + = = + + = + + ω ω ω φ ω

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2 2 1 < L R LC 2 2 1 > L R LC 2 2 1 = L R LC