ch31-p021 - 21. (a) The charge (as a function of time) is...

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Unformatted text preview: 21. (a) The charge (as a function of time) is given by q = Q sin ωt , where Q is the maximum charge on the capacitor and ω is the angular frequency of oscillation. A sine function was chosen so that q = 0 at time t = 0. The current (as a function of time) is dq = ωQ cos ωt , dt i= and at t = 0, it is I = ωQ. Since ω = 1/ LC , b Q = I LC = 2.00 A . g c3.00 × 10 Hhc2.70 × 10 Fh = 180 × 10 −3 −6 −4 C. (b) The energy stored in the capacitor is given by UE = q 2 Q 2 sin 2 ωt = 2C 2C and its rate of change is dU E Q 2ω sin ωt cos ωt = dt C bg 1 We use the trigonometric identity cosωt sin ωt = 2 sin 2ωt to write this as dU E ωQ 2 sin 2ωt . = dt 2C bg The greatest rate of change occurs when sin(2ωt) = 1 or 2ωt = π/2 rad. This means t= ππ π = LC = 4ω 4 4 ( 3.00 ×10 −3 H )( 2.70 ×10−6 F ) = 7.07 ×10−5 s. (c) Substituting ω = 2π/T and sin(2ωt) = 1 into dUE/dt = (ωQ2/2C) sin(2ωt), we obtain FG dU IJ H dt K E max 2 πQ 2 πQ 2 = = . TC 2TC c3.00 × 10 Hhc2.70 × 10 Fh = 5.655 × 10 −3 Now T = 2 π LC = 2 π FG dU IJ H dt K = E max −6 c π 180 × 10−4 C . h −4 s, so 2 c5.655 × 10 shc2.70 × 10 Fh = 66.7 W. −4 −6 We note that this is a positive result, indicating that the energy in the capacitor is indeed increasing at t = T/8. ...
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This note was uploaded on 06/03/2011 for the course PHY 2049 taught by Professor Any during the Spring '08 term at University of Florida.

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