 # 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

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.

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