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ISM_CH31

ISM_CH31 - Chapter 31 1(a All the energy in the circuit...

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Chapter 31 1. (a) All the energy in the circuit resides in the capacitor when it has its maximum charge. The current is then zero. If Q is the maximum charge on the capacitor, then the total energy is U Q C = = × × = × - - - 2 6 2 6 6 2 2 90 10 2 360 10 117 10 . . . C F J. c h c h (b) When the capacitor is fully discharged, the current is a maximum and all the energy resides in the inductor. If I is the maximum current, then U = LI 2 /2 leads to I U L = = × × = × - - - 2 2 1168 10 75 10 558 10 6 3 3 . . J H A. c h 2. According to U LI Q C = = 1 2 2 1 2 2 , the current amplitude is I Q LC = = × × × = × - - - - 300 10 4 00 10 4 52 10 6 3 6 2 . . . C 1.10 10 H F A. c hc h 3. We find the capacitance from U Q C = 1 2 2 : C Q U = = × × = × - - - 2 6 2 6 9 2 160 10 2 140 10 914 10 . . C J F. c h c h 4. (a) The period is T = 4(1.50 μ s) = 6.00 μ s. (b) The frequency is the reciprocal of the period: f T = = = × 1 1 6 00 167 10 5 . . μ s Hz. (c) The magnetic energy does not depend on the direction of the current (since U B i 2 ), so this will occur after one-half of a period, or 3.00 μ s. 1241

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CHAPTER 31 5. (a) We recall the fact that the period is the reciprocal of the frequency. It is helpful to refer also to Fig. 31-1. The values of t when plate A will again have maximum positive charge are multiples of the period: t nT n f n n A = = = × = 2 00 10 500 3 . . , Hz s μ b g where n = 1, 2, 3, 4, K . The earliest time is ( n =1) 5.00 s. A t μ = (b) We note that it takes t T = 1 2 for the charge on the other plate to reach its maximum positive value for the first time (compare steps a and e in Fig. 31-1). This is when plate A acquires its most negative charge. From that time onward, this situation will repeat once every period. Consequently, ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 3 2 1 2 1 1 1 ( 1) 2 1 2 1 2.50 s , 2 2 2 2 2 10 Hz n n t T n T n T n f μ - - = + - = - = = = - × where n = 1, 2, 3, 4, K . The earliest time is ( n =1) 2.50 s. t μ = (c) At t T = 1 4 , the current and the magnetic field in the inductor reach maximum values for the first time (compare steps a and c in Fig. 31-1). Later this will repeat every half- period (compare steps c and g in Fig. 31-1). Therefore, ( 29 ( 29 ( 29 ( 1) 2 1 2 1 1.25 s , 4 2 4 L T n T T t n n μ - = + = - = - where n = 1, 2, 3, 4, K . The earliest time is ( n =1) 1.25 s. t μ = 6. (a) The angular frequency is ϖ = = = × = - k m F x m 8 0 050 89 13 . . . N 2.0 10 m kg rad s c hb g (b) The period is 1/ f and f = ϖ /2 π . Therefore, T = = = × - 2 2 7 0 10 2 π π 89 ϖ rad s s. . (c) From ϖ = ( LC ) –1/2 , we obtain 218
C L = = = × - 1 1 89 50 2 5 10 2 2 5 ϖ rad s H F. b gb g . . 7. (a) The mass m corresponds to the inductance, so m = 1.25 kg. (b) The spring constant k corresponds to the reciprocal of the capacitance. Since the total energy is given by U = Q 2 /2 C , where Q is the maximum charge on the capacitor and C is the capacitance, C Q U = = × × = × - - - 2 6 2 6 3 2 175 10 2 570 10 2 69 10 C J F c h c h . . and k = × = - 1 2 69 10 372 3 . m / N N / m. (c) The maximum displacement corresponds to the maximum charge, so 4 max 1.75 10 m. x - = × (d) The maximum speed v max corresponds to the maximum current. The maximum current is I Q Q LC = = = × × = × - - - ϖ 175 10 125 2 69 10 302 10 6 3 3 C H A.

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