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HW-18Solutions-03-12-08 - Lehigh University HW-18 Solutions...

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1 Lehigh University Physics 21, Spring 2008 March 12, 2008 HW-18 Solutions 18-1. (HRW 31-24) In an oscillating series RLC circuit, find the time required for the maximum energy present in the capacitor during an oscillation to fall to half its initial value. Assume at . Solution: 24. The assumption stated at the end of the problem is equivalent to setting φ = 0 in Eq. 31-25. Since the maximum energy in the capacitor (each cycle) is given by q C max / 2 2 , where q max is the maximum charge (during a given cycle), then we seek the time for which 2 2 max max 1 . 2 2 2 2 q Q Q q C C = = Now q max (referred to as the exponentially decaying amplitude in §31-5) is related to Q (and the other parameters of the circuit) by q Qe q Q Rt L Rt L max / max ln . = F H G I K J = − 2 2 Setting q Q max = / 2 , we solve for t : t L R q Q L R L R = − F H G I K J = − F H G I K J = 2 2 1 2 2 ln ln ln . max The identities ln( / ) ln ln 1 2 2 2 1 2 = − = − were used to obtain the final form of the result. 18-2. (HRW 31-29) (a) At what frequency would a 6.0 mH inductor and a 10 μ F capacitor have the same reactance? (b) What would the reactance be? (c) Show that this frequency would be the natural frequency of an oscillating circuit with the same L and C.
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