Chapter 31a

# Chapter 31a - Alternating current(AC circuits(Chapt 31 VC I...

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Alternating current (AC) circuits (Chapt. 31) t v i T 2T V C –V C v, i I –I

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The LC oscillator Consider the experiment of charging a capacitor to a potential V C and now discharging it across a resistor , v(t) ( convention: time varying quantities: lower case symbols) As the capacitor discharges, the voltage across the resistor (and capacitor) decay monotonically as, t RC C vV e = Since for the resistor v vi R i R = →= we similarly have that, t C RC V ie R = C R ( lower case time varying) ( lower case time varying)
t RC C v(t) V e = t C RC V i(t) e R = C V C V R v(t) i (t) time, t 0 v, i These are both simple monotonic decays: v(t) C R

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We ask now what happens if we instead place an inductor across the charged capacitor? (The behavior is very different ). The capacitor initially has charge ± Q o on its plates and potential V C . +Q o –Q o a The loop rule holds at each instant in time so starting at the point marked a and going around the circuit gives, CL vv0 −= The emf induced across the inductor, v L comes in with the minus sign because Lenz’s law tells us that it must oppose changes in the quantities associated with it. The LC circuit L di vL dt = where, (self inductance) C L
But for a capacitor, CC q qC v v C =→ = so, di q L0 dt C −+ = C di Lv 0 dt = di q 0 dt LC = But, dq i dt =− – sign because the capacitor’s charge decreases while its current increases & visa versa. dd q q 0 dt dt LC ⎛⎞ −− + = ⎜⎟ ⎝⎠ So, or, Rearranging, C L

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Giving the differential equation, 2 2 dq 1 q0 LC dt + = This has the form of the diff. eq. of motion for the simple harmonic oscillator, 2 2 dx k x0 m dt + = Which has general solution, o x(t) X cos( t ) = ω+φ where, k m ω= X o is the amplitude and φ is a phase constant determined by the initial conditions (i.e. if φ = 0, the mass had its positive extreme value of x = X o , when t = 0). By analogy, we have for the LC circuit the general solution, o qQ c o s ( t ) + φ where, 1 is the natural (angular) frequency of oscillation C L
C L Unlike the RC circuit which merely decays with time, the charge on the capacitor in the LC circuit oscillates harmonically in time ! o qQ c o s ( t ) + φ To get the current we differentiate this wrt time, o dq d i[ Q c o s ( t ) ] dt dt == ω + φ o iQ s i n ( t ) =− ω ω +φ Designating, o IQ iI s i n ( t ) =− The potential across the capacitor is given by recalling that q = Cv C o C Q q vc o s ( t ) CC = + φ 1 LC ω= , (the same instantaneous i everywhere)

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C L CL vv0 = o LC Q vv c o s ( t ) C == ω + φ = So both the voltage and the current oscillate in time. This same voltage must appear across the inductor since the loop rule gave that, = To consider a specific case of this general solution take t = 0 as the instant at which the inductor is connected. At that time (t = 0) the capacitor has its full charge so, oC QC V = o C Q V C = At the instant we hook–up the inductor we must then have that oo CC QQ v( t 0 ) V c o s (t ) = = ω + φ 0 φ =
So in this case, C qC V c o s ( t ) iI s i n ( t ) =− ω C vV c o s ( t ) 1 LC ω= The period, T is, 2 T π = ω T2 L C t v i T 2T V C –V C v, i I –I C IC V = ω C L

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HRW derives these expressions from energy considerations (read there).
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Chapter 31a - Alternating current(AC circuits(Chapt 31 VC I...

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