1018 CHAPTER 30 Inductance reading of each meter after a long time has ela psed. (c ) Find the maximum charge on the capacitor. (d) Draw a qualitative graph of the reading of voltmeter as a function of time. 30.66 . In the circuit shown in Fig. P30. 66 the bat tery an d the ind ucto r have no appreciable internal resist- ance and there is no current in the cir- cuit. After the switch is closed, find the readings of the ammeter (A) and voltmeters and (a) the instant after the switch is closed and (b) after the switch has been closed for a very long time. (c) Which answers in parts (a) and (b) would change if the inductance were 24.0 mH instead? 30.67 .. CP In the circuit shown in Fig. P30.67, switch S is closed at time (a) Find the reading of each meter just after S is closed. (b) What does each meter read long after S is closed? 30.68 .. In the circuit shown in Fig. P30.68, switch S 1 has been closed for a long enough time so that the current reads a steady 3.50 A. Suddenly, switch is closed and is opened at the same instant. (a) What is the maximum charge that the capacitor will receive? (b) What is the current in the inductor at this time? 30.69 .. CP In the circuit shown in Fig. P30.69, and Switch S is closed at Just after the switch is closed, (a) what is the potential difference across the resistor (b) which point, a or b, is at a higher potential; (c) what is the R 1 ; v ab t = 0. L = 0. 30 0 H. R 2 = 25.0 Æ, 40.0 Æ, R 1 = E = 60 .0 V, S 1 S 2 t = 0. V 2 2 1V 1 V 2 potential difference across the inductor L; (d) which point, c or d , is at a higher potential? The switch is left closed a long time and then opened. Just after the switch is opened, (e) what is the potential difference across the resistor (f) which point, a or b, is at a higher potential; (g) what is the potential difference across the inductor L; (h) which point, c or d , is at a higher potential? 30.70 .. CP In the circuit shown in Fig. P30.69, and (a) Switch S is closed. At some time t afterward, the current in the inductor is increasing at a rate of At this instant, what are the current through and the current through ( Hint: Analyze two separate loops: one containing and and the other containing and L.) (b) After the switch has been closed a long time, it is opened again. Just after it is opened, what is the current through 30.71 .. CALC Consider the circuit shown in Fig. P30.71. Let and (a) Switch is closed and switch is left open. Just after is closed, what are the current through and the potential differences and (b) After has been closed a long time is still open) so that the current has reached its final, steady value, what are and (c) Find the expressions for and as functions of the time t since was closed. Y our results should agree with part (a) when and with part (b) when Graph and versus time. 30.72 .. After the current in the cir cuit of Fig. P30.71 has reached its final, steady value with switch closed and open, switch is closed, thus short-circuiting the inductor. (Switch remains closed. See Problem 30.71 for numerical values of the circuit ele- ments.) (a) Just after is closed, what are and and what are the currents through R, and (b) A long time after is closed, what are and and what are the currents through R, and (c) Derive expressions for the currents through R, and as functions of the time t that has elapsed since was closed. Y our results should agree with part (a) when and with part (b) when Graph these three currents versus time. 30.73 ... CP CALC We have ig- nore d the va riati on of the ma g- netic field across the cross section of a toroidal solenoid. Let’s now examine the validity of that ap- prox imat ion. A certa in toro idal solenoid has a rectangular cross section (Fig. P30.73). It has N uniformly spaced turns, with air inside. The magnetic field at a point inside the toroid is given by the equation derived in Example 28.10 (Section 28.7). Do not assume the field is uniform over the cross section. (a) Show that the magnetic flux through a cross section of the toroid is (b) Show that the inductance of the toroidal solenoid is given by L = m 0 N 2 h 2p ln a b a b £ B = m 0 Nih 2p ln a b a b t S q. t = 0 S 2 S 2 R 0 , S 2 ? R 0 , v cb , v ac S 2 S 2 ? R 0 , v cb , v ac S 2 S 1 S 2 S 2 S 1 v cb v ac , i 0 , t S q. t = 0 S 1 v cb v ac , i 0 , v cb ? v ac , i 0 , 1S 2 S 1 v cb ? v ac R 0 i 0 S 1 S 2 S 1 4. 00 H. L = R = 150 Æ, R 0 = 50.0 Æ, E = 36 .0 V, R 1 ? E , R 2 , R 1 E R 2 ? i 2 R 1 i 1 di> dt = 50.0 A> s. L = 0. 30 0 H. R 2 = 25.0 Æ, 40.0 Æ, R 1 = E = 60 .0 V, v cd R 1 ; v ab v cd S V 2 V 3 V 1 V 4 A 2 V 5 A 4 A 3 A 1 50.0 V 50.0 V 100.0 V 5.00 mH 40.0 V 12.0 mF + Figure P30.65 12.0 mH V 2 V 1 S 15.0 V A 25.0 V + Figure P30.66 S 15.0 V 40.0 V 5.0 V 10.0 V 20.0 mH 10.0 mH 25.0 V A 1 A 2 A 3 A 4 Figure P30.67 + 2.0 mH 5.0 mF S 1 R S 2 E A Figure P30.68 S R 2 L R 1 a b c d E Figure P30.69 + S 2 S 1 R 0 a c R L b E Figure P30.71 a r b h Figure P30.73