Notes on Chapter 31 - 31-1 Electromagnetic Oscillations &...

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13 July 2010 1 Electromagnetic Oscillations & Alternating Currents In this chapter we consider: 1. Oscillations in LC circuits 2. Damped oscillations in LRC circuits 3. Alternating currents (ac), and 4. Transformers 31-1
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13 July 2010 2 Recall from mechanics – conservation of energy
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13 July 2010 3 In RC and RL circuits charge q(t) and current i(t) and potential difference v(t) grow and decay exponentially In an LC circuit, the charge, current and potential difference do not decay. Instead they vary sinusoidally electromagnetic oscillations
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13 July 2010 4 2 Energy stored in electric field of capacit r: 2 o E U q C = 2 Energy stored in magnetic field of induct r: o 2 B Li U = represent instantaneous q () , () , uantities represent amplitudes it qt vt IQV LC Oscillations (Qualitative) 1 C v qt C  =   R v itR = 31-2
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13 July 2010 5 The Electromagnetic- Mechanical Analogy Mass-Spring System LC Oscillator x q v = dx/dt i = dq /dt k 1/C m L U = ½kx 2 U E = ½(1/C)q 2 K = ½mv 2 U B = ½Li 2 / km ω= 1/ LC 31-3
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13 July 2010 6 LC Oscillations (Quantitatively) 22 11 () Constant () () () () t E t U t K t k x t mv = = += + 0 dE d kx mv dt dt dv dx mv kx dt dt    = + = 2 2 0 dx m kx dt cos( ) xX t ωφ = Revisit Mass-Spring System Used: dx v dt = 31-4 sin( ) vX t ω = +
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13 July 2010 7 LC Oscillator 1 22 11 () Constant EB C tt E t U U q Li = = += + 0 q C dE d q Li C dt dt dq di Li dt dt    = = 2 2 1 0 C dq Lq dt sin( ) sin( ) cos( ) dq Qt It dt qQ t i ω φ ωφ = + = + = Used: dq i dt =
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13 July 2010 8 The instantaneous values of the electrical and magnetic energies are: 22 2 cos ( ) E qQ Ut CC ωφ = = 1 2 2 2 2 1 2 1 2 [ sin( )] sin ( ) sin ( ) 2 B U Li L Q t LQ t Q t C ω = = −+ = + = 2 2 () cos ( ) 2 E t Q C = 2 2 sin ( ) 2 B t Q C = 0 φ=
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13 July 2010 9 Damped Oscillations in an RLC Circuit d F bv = − Revisit damped harmonic motion (PHYS191) bv kx ma mx −− = =  2 2 0 d x dx m b kx dt dt + += ( /2 ) ( ) x cos( ) bt m m xt e t ωφ = + 2 2 4 kb m m ω = − 31-5 http://www.lon-capa.org/~mmp/applist/damped/d.htm
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13 July 2010 1 22 11 EB C E U U q Li = += + The total electromagnetic energy is: The total energy decreases as electromagnetic energy is transferred to thermal energy at a rate given by: 2 di q dq dt C dt dE i R Li dt + = −= 2 2 2 0 di q dq dt C dt d q dq q R dt C dt Li i R L ++ = ( /2 ) ( ) cos( ) Rt L q t Qe t ωφ = + 2 2 ( /2 ) 1 4 RL R LC L ω = − = − /2 cos ( ) Rt L E qQ Uet CC
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This note was uploaded on 09/30/2010 for the course PHYS MTH 203 taught by Professor None during the Spring '10 term at American University of Sharjah.

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Notes on Chapter 31 - 31-1 Electromagnetic Oscillations &...

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