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1. The current in a coil changes from Ii to If in the same direction in time t. If the average emf induced in the coil is , what is the inductance of the coil? L | I Ii | dI | | L f t dt | | t L | If Ii | 2. A technician wraps wire around a tube of length having a diameter of D. When the windings are evenly spread over the full length of the tube, the result is a solenoid containing N turns of wire. (a) Find the inductance of this solenoid. (b) If the current in this solenoid increases at the rate of r, find the self-induced emf in the solenoid? (a) For a solenoid, B 0 nI B NBA 0 nNIA d d d B - 0 nNIA - 0 nNA I dt dt dt d By definition, -L I dt L 0 nNA - D 1 N But A ( ) 2 D 2 and n 2 4 N 1 L 0 ( )N( D 2 ) 0 N 2 D 2 4 4 (b) -L d d I | |L| I| dt dt | | Lr | | 0 4 N2D2r 3. A N turn solenoid has a radius of r and an overall length of . (a) What is its inductance? (b) If the solenoid is connected in series with a resistor R and a battery, what is the time constant of the circuit? (a) For a solenoid, B 0 nI B NBA 0 nNIA d d d B - 0 nNIA - 0 nNA I dt dt dt d By definition, -L I dt L 0 nNA - But A r 2 and n N N L 0 ( )N(r 2 ) 0 N 2 r 2 0 L (b) R N2r 2 R 0 R N2r 2 4. A battery of emf V is connected into a series circuit containing a resistor R and an inductor L. (a) In what time interval will the current reach 50.0% of its final value? (b) In what time interval will the current reach 90.0% of its final value? L Solution V I V (1 - e - t/ ) R R L R V when t R 1V (a) We want to find t when I 2R 1V V 1 (1 - e - t/ ) 1 - e - t/ 2R R 2 1 e - t/ 2 t 1 n 2 t n2 Final value of I t n2 L t n2 R (b) We want to find t when I 9 V 10 R 9 V V 9 (1 - e - t/ ) 1 - e - t/ 10 10 R R 1 e - t/ 10 t 1 n 10 t n10 t n10 t L n10 R 5. Consider the circuit shown in the figure. V (a) When the switch is in position a, for what value of R1 will the circuit have a time constant of ? (b) What is the current in the inductor at the instant the switch is thrown to position b? (a) L L L R 1R 2 R R 1 // R 2 R1 R 2 L(R 1 R 2 ) R 1R 2 R 1 R 2 L(R 1 R 2 ) R 1 (R 2 L) LR 2 R1 LR 2 R 2 L (b) Current in the inductor at the instant the switch is thrown to position b Current in the inductor before the switch is thrown to position b V R 1 // R 2 L R 1R 2 R1 R 2 V(R 1 R 2 ) R 1R 2 6. Consider the circuit in the figure below. (a) What is the inductive time constant of the circuit? Ans. =L/R (b) Calculate the current in the circuit t after the switch is closed. I R - t (1 e ) R (1 e - Rt L ) (c) What is the value of the final steady-state current? Ans. Final steady-state current = /R (d) After what time interval does the current reach a factor f of its maximum value? I - t - t (1 e ) I max (1 e ) where I max R When I fI max - t fI max I max (1 e ) f 1 e e - t t 1 f t n(1 f) 1 ) 1- f 1 t n( ) 1- f L 1 t n( ) R 1- f t - - n( R 7. A inductor L and a resistor R are connected with a switch to a battery as shown in the figure below. (a) After the switch is first thrown to a (connecting the battery), what time interval elapses before the current reaches I? I e - Rt L R - (1 e ) 1 t R (1 e - Rt L ) RI RI RI Rt n( ) L Rt n( ) RI L L t n( ) RI R e Rt L (b) What is the current in the inductor t after the switch is closed? I - t - Rt L (1 e ) (1 e ) R R (c) Now the switch is quickly thrown from a to b. What time interval elapses before the current in the inductor falls to I? I e Rt L R - t e R RI Rt RI n( ) L Rt n( ) L RI L t n( ) R RI - e - Rt L 8. An air-core solenoid with N turns length and has a diameter of D. When the solenoid carries a current of I, how much energy is stored in its magnetic field? For a solenoid, B 0 nI B NBA 0 nNIA d d d B - 0 nNIA - 0 nNA I dt dt dt d By definition, -L I dt L 0 nNA - D 1 But A ( ) 2 D 2 2 4 N 1 L 0 ( )N( D 2 ) 0 N 2 D 2 4 4 1 Energy stored LI 2 2 1 ( 0 N 2 D 2 )I 2 2 4 0 8 N2D2I2 9. A battery V, a resistor R, and a inductor L are connected in series. After the current in the circuit has reached its maximum value, calculate the following. (a) the power being supplied by the battery (b) the power being delivered to the resistor (c) the power being delivered to the inductor (d) the energy stored in the magnetic field of the inductor (a) Power supplied by the battery = power dissipated by R = V2/R (b) Power dissipated by R = V2/R (c) At equilibrium, potential difference across the inductor = 0 Power delivered to the inductor = 0 (d) Energy stored in the magnetic field = 1 2 1 V 2 LV 2 LI L( ) 2 2 R 2R 10. The magnetic field inside a superconducting solenoid is B. The solenoid has an inner diameter of D and a length of . (a) Determine the magnetic energy density in the field. (b) Determine the energy stored in the magnetic field within the solenoid. (a) Energy density 1 B2 2 0 (b) Energy stored Energy density Volume 1 D B2 ( ) 2 2 0 2 2 2 B D 8 0 11. In the circuit of Figure P32.48, the battery emf is , the resistance is R, and the capacitance is C. The switch S is closed for a long time, and no voltage is measured across the capacitor. After the switch is opened, the potential difference across the capacitor reaches a maximum value of V. What is the value of the inductance L? Figure P32.48 1 Just before the switch is opened, energy stored in inductor LI 2 2 1 L( ) 2 2 R After switched is closed : 1 2 L( ) Energy stored in inductor energy stored in capacitor 2 R When the potential difference across the capacitor is maximum, energy stored in capacitor is maximum and energy stored in the inductor is 0. 1 2 1 L( ) 0 CV 2 L( ) 2 CV 2 2 R 2 R R L CV 2 ( ) 2 L ( VR )2 C 12. Calculate the inductance of an LC circuit that oscillates at frequency f when the capacitance is C. 0 1 LC 0 2 f 2 f 1 LC 1 4 2 f 2 LC 1 L 2 2 4 f C 13. LC circuit consists of an inductor L and a capacitor C. If the maximum instantaneous current is I, what is the greatest potential difference across the capacitor? 1 Energy stored in the circuit LI 2 2 When the potential difference across the capacitor is maximum, energy stored in capacitor is maximum and energy stored in the inductor is 0. 1 1 LI 2 0 CV 2 LI 2 CV 2 2 2 1 L CV 2 2 I V L ( )2 C I 14. The switch in the figure below is connected to position a for a long time interval. At t = 0, the switch is thrown to position b. After this time, what are the following? R L (a) the frequency of oscillation of the LC circuit. (b) the maximum charge that appears on the capacitor. (c) the maximum current in the inductor. (d) the total energy the circuit possesses at t . (a) 1 LC 2f f 1 f 2 2 LC (b) Maximum charge = charge in C just before the switch is thrown to position b = C (c) Consivation of energy : 1 Q 2max 1 2 Q 2max 2 LImax I max 2 C 2 LC Q C C I max max or L LC LC (d) Consivation of energy Total energy is constant, independent of time. 1 Q 2max 1 (C ) 2 1 U C 2 2 C 2 C 2 15. An LC circuit like the one in the figure below contains an inductor L and a capacitor C that initially carries a charge Qmax. The switch is open for t < 0 and is then thrown closed at t = 0. (a) Find the frequency (in hertz) of the resulting oscillations. (b) Find the charge on the capacitor at t. (c) Find the current in the circuit at t. (a) 1 LC 2f f 1 f 2 2 LC (b) Since Q Q max at t 0 and it is going to decease in the near future t. Q Q max cos t with 1 LC t Q max cos LC (c) Q Q max cos t t LC d d Q [Q max cos dt dt t LC ] Q max - Q max LC d cos dt sin t LC t LC