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Chapter 11 Review

Key Terms

henry (H)
unit of inductance,

; it is also expressed as a volt second per ampere



inductance
property of a device that tells how effectively it induces an emf in another device



inductive time constant
denoted by

, the characteristic time given by quantity

of a particular series

circuit



inductor
part of an electrical circuit to provide self-inductance, which is symbolized by a coil of wire



LC circuit
circuit composed of an ac source, inductor, and capacitor



magnetic energy density
energy stored per volume in a magnetic field



mutual inductance
geometric quantity that expresses how effective two devices are at inducing emfs in one another



RLC circuit
circuit with an ac source, resistor, inductor, and capacitor all in series.



self-inductance
effect of the device inducing emf in itself



Key Equations


Mutual inductance by flux
Mutual inductance in circuits
Self-inductance in terms of magnetic flux
Self-inductance in terms of emf
Self-inductance of a solenoid
Self-inductance of a toroid
Energy stored in an inductor
Current as a function of time for a RL circuit
Time constant for a RL circuit
Charge oscillation in LC circuits
Angular frequency in LC circuits
Current oscillations in LC circuits
Charge as a function of time in RLC circuit
Angular frequency in RLC circuit



Summary

11.1 Mutual Inductance

  • Inductance is the property of a device that expresses how effectively it induces an emf in another device.
  • Mutual inductance is the effect of two devices inducing emfs in each other.
  • A change in current in one circuit induces an emf in the second:
  • where is defined to be the mutual inductance between the two circuits and the minus sign is due to Lenz’s law.
  • Symmetrically, a change in current through the second circuit induces an emf in the first:

where

is the same mutual inductance as in the reverse process.



11.2 Self-Inductance and Inductors

  • Current changes in a device induce an emf in the device itself, called self-inductance,
  • where is the self-inductance of the inductor and is the rate of change of current through it. The minus sign indicates that emf opposes the change in current, as required by Lenz’s law. The unit of self-inductance and inductance is the henry (), where
  • The self-inductance of a solenoid is
  • where is its number of turns in the solenoid, is its cross-sectional area, is its length, and is the permeability of free space.
  • The self-inductance of a toroid is
  • where is its number of turns in the toroid, and are the inner and outer radii of the toroid, is the height of the toroid, and is the permeability of free space.


11.3 Energy in a Magnetic Field

  • The energy stored in an inductor is
  • The self-inductance per unit length of coaxial cable is



11.4 RL Circuits

  • When a series connection of a resistor and an inductor—an circuit—is connected to a voltage source, the time variation of the current is
  • (turning on),where the initial current is
  • The characteristic time constant is where is the inductance and is the resistance.
  • In the first time constant the current rises from zero to and to of the remainder in every subsequent time interval
  • When the inductor is shorted through a resistor, current decreases as



Current falls to

in the first time interval

and to

of the remainder toward zero in each subsequent time



11.5 Oscillations in an LC Circuit

  • The energy transferred in an oscillatory manner between the capacitor and inductor in an circuit occurs at an angular frequency
  • The charge and current in the circuit are given by



11.6 RLC Series Circuits

  • The underdamped solution for the capacitor charge in an circuit is

The angular frequency given in the underdamped solution for the

circuit is



Answers to Check Your Understanding

11.1



11.2 a. decreasing; b. increasing; Since the current flows in the opposite direction of the diagram, in order to get a positive emf on the left-hand side of diagram (a), we need to decrease the current to the left, which creates a reinforced emf where the positive end is on the left-hand side. To get a positive emf on the right-hand side of diagram (b), we need to increase the current to the left, which creates a reinforced emf where the positive end is on the right-hand side.



11.3



11.4 a.

; b.



11.5 a.

b.



11.6

11.8 a.

; b.

; c.



11.10 a.

; b.

or

; c.



11.11 a. overdamped; b.



Conceptual Questions

11.1 Mutual Inductance

1. Show that

and

which are both expressions for self-inductance, have the same units.



2. A

inductor carries a current of

Describe how a

emf can be induced across it.



3. The ignition circuit of an automobile is powered by a

battery. How are we able to generate large voltages with this power source?



4. When the current through a large inductor is interrupted with a switch, an arc appears across the open terminals of the switch. Explain.



11.2 Self-Inductance and Inductors

5. Does self-inductance depend on the value of the magnetic flux? Does it depend on the current through the wire? Correlate your answers with the equation



6. Would the self-inductance of a

long, tightly wound solenoid differ from the self-inductance per meter of an infinite, but otherwise identical, solenoid?



7. Discuss how you might determine the self-inductance per unit length of a long, straight wire.



8. The self-inductance of a coil is zero if there is no current passing through the windings. True or false?



9. How does the self-inductance per unit length near the centre of a solenoid (away from the ends) compare with its value near the end of the solenoid?



11.3 Energy in a Magnetic Field

10. Show that

has units of energy.



11.4 RL Circuits

11. Use Lenz’s law to explain why the initial current in the

circuit of Figure 11.4.1(b) is zero.



12. When the current in the

circuit of Figure 11.4.1(b) reaches its final value

what is the voltage across the inductor? Across the resistor?



13. Does the time required for the current in an

circuit to reach any fraction of its steady-state value depend on the emf of the battery?



14. An inductor is connected across the terminals of a battery. Does the current that eventually flows through the inductor depend on the internal resistance of the battery? Does the time required for the current to reach its final value depend on this resistance?



15. At what time is the voltage across the inductor of the

circuit of Figure 14.12(b) a maximum?



16. In the simple

circuit of Figure 11.4.1(b), can the emf induced across the inductor ever be greater than the emf of the battery used to produce the current?



17. If the emf of the battery of Figure 11.4.1(b) is reduced by a factor of

by how much does the steady-state energy stored in the magnetic field of the inductor change?



18. A steady current flows through a circuit with a large inductive time constant. When a switch in the circuit is opened, a large spark occurs across the terminals of the switch. Explain.



19. Describe how the currents through

and

shown below vary with time after switch

is closed.



20. Discuss possible practical applications of

circuits.



11.5 Oscillations in an LC Circuit

21. Do Kirchhoff’s rules apply to circuits that contain inductors and capacitors?



22. Can a circuit element have both capacitance and inductance?



23. In an

circuit, what determines the frequency and the amplitude of the energy oscillations in either the inductor or capacitor?



11.6 RLC Series Circuits

24. When a wire is connected between the two ends of a solenoid, the resulting circuit can oscillate like an

circuit. Describe what causes the capacitance in this circuit.



25. Describe what effect the resistance of the connecting wires has on an oscillating

circuit.



26. Suppose you wanted to design an

circuit with a frequency of

What problems might you encounter?



27. A radio receiver uses an

circuit to pick out particular frequencies to listen to in your house or car without hearing other unwanted frequencies. How would someone design such a circuit?



Problems

11.1 Mutual Inductance

28. When the current in one coil changes at a rate of

an emf of

is induced in a second, nearby coil. What is the mutual inductance of the two coils?



29. An emf of

is induced in a coil while the current in a nearby coil is decreasing at a rate of

What is the mutual inductance of the two coils?



30. Two coils close to each other have a mutual inductance of

If the current in one coil decays according to

where

and

what is the emf induced in the second coil immediately after the current starts to decay? At

?



31. A coil of

is wrapped around a long solenoid of cross-sectional area

The solenoid is

long and has

(a) What is the mutual inductance of this system? (b) The outer coil is replaced by a coil of

whose radius is three times that of the solenoid. What is the mutual inductance of this configuration?



32. A

solenoid is

long and

in diameter. Inside the solenoid, a small

single-turn rectangular coil is fixed in place with its face perpendicular to the long axis of the solenoid. What is the mutual inductance of this system?



33. A toroidal coil has a mean radius of

and a cross-sectional area of

; it is wound uniformly with

A second toroidal coil of

is wound uniformly over the first coil. Ignoring the variation of the magnetic field within a toroid, determine the mutual inductance of the two coils.



34. A solenoid of

turns has length

and radius

and a second smaller solenoid of

turns has length

and radius

The smaller solenoid is placed completely inside the larger solenoid so that their long axes coincide. What is the mutual inductance of the two solenoids?



11.2 Self-Inductance and Inductors

35. An emf of

is induced across a coil when the current through it changes uniformly from

to

in

What is the self-inductance of the coil?



36. The current shown in part (a) below is increasing, whereas that shown in part (b) is decreasing. In each case, determine which end of the inductor is at the higher potential.



37. What is the rate at which the current though a

coil is changing if an emf of

is induced across the coil?



38. When a camera uses a flash, a fully charged capacitor discharges through an inductor. In what time must the

current through a

inductor be switched on or off to induce a

emf?



39. A coil with a self-inductance of

carries a current that varies with time according to

Find an expression for the emf induced in the coil.



40. A solenoid

long is wound with

of wire. The cross-sectional area of the coil is

What is the self-inductance of the solenoid?



41. A coil with a self-inductance of

carries a current that decreases at a uniform rate

What is the emf induced in the coil? Describe the polarity of the induced emf.



42. The current

through a

inductor varies with time, as shown below. The resistance of the inductor is

Calculate the voltage across the inductor at

and



43. A long, cylindrical solenoid with

has a radius of

(a) Neglecting end effects, what is the self-inductance per unit length of the solenoid? (b) If the current through the solenoid changes at the rate

what is the emf induced per unit length?



44. Suppose that a rectangular toroid has

windings and a self-inductance of

If

what is the ratio of its outer radius to its inner radius?



45. What is the self-inductance per meter of a coaxial cable whose inner radius is

and whose outer radius is

?



11.3 Energy in a Magnetic Field

46. At the instant a current of

is flowing through a coil of wire, the energy stored in its magnetic field is

What is the self-inductance of the coil?



47. Suppose that a rectangular toroid has

windings and a self-inductance of

If

what is the current flowing through a rectangular toroid when the energy in its magnetic field is

?



48. Solenoid

is tightly wound while solenoid

has windings that are evenly spaced with a gap equal to the diameter of the wire. The solenoids are otherwise identical. Determine the ratio of the energies stored per unit length of these solenoids when the same current flows through each.



49. A

inductor carries a current of

How much ice at

could be melted by the energy stored in the magnetic field of the inductor? (Hint: Use the value

for ice.)



50. A coil with a self-inductance of

and a resistance of

carries a steady current of

(a) What is the energy stored in the magnetic field of the coil? (b) What is the energy per second dissipated in the resistance of the coil?



51. A current of

is flowing in a coaxial cable whose outer radius is five times its inner radius. What is the magnetic field energy stored in a

length of the cable?



11.4 RL Circuits

52. In Figure 11.4.1,

and

Determine (a) the time constant of the circuit, (b) the initial current through the resistor, (c) the final current through the resistor, (d) the current through the resistor when

and (e) the voltages across the inductor and the resistor when



53. For the circuit shown below,

and

After steady state is reached with