Chapter 24 Inductance and Circuit Oscillations

Main Points

• Inductance and Inductors

• Energy in Inductors and in the Magnetic Field

• RL and RC Circuits

• LC Circuits

• RLC Circuits, Damped Oscillations

• Faraday’s Law: Changing current in a circuit will induce emf in that circuit as well as others nearby

24-1 Inductance and Inductor

Self-Inductance: Circuit induces emf in itself

Mutual Inductance: Circuit induces emf in second circuit

Self Inductance

Switch closes

Self-Induction: changing current through a loop inducing an opposing emf in that same loop.

• The flux, therefore, is also proportional to the current.

I• The magnetic field produced by

the current in the loop shown is proportional to that current:

• We define this constant of proportionality between flux and current to be the inductance, L.

LIB

• Combining with Faraday’s Law gives the emf induced by a changing current:

Self-induced emf

The minus sign indicates that—as the law states—the self-induced emf   has the orientation such that it opposes the change in current I

The inductance L is a proportionality constant that depends on the geometry of the circuit

SI :H (Wb/A)

(a) The current i is increasing and the self-induced appears along the coil in a direction such that it opposes the increase.

(b) The current i is decreasing and the self-induced emf appears in a direction such that it opposes the decrease.

• The inductance of an inductor (a set of coils in some geometry; e.g., solenoid) then, can be calculated from its geometry alone if the device is constructed from conductors and air (similar to the capacitance of a capacitor).

• If extra material (e.g., an iron core) is added, the inductance will increase (just as adding a dielectric increases capacitance)

• Archetypal inductor is a long solenoid, just as a pair of parallel plates is the archetypal capacitor.

d

A

- - - - -

+ + + +r << l

l

r

N turns

B

The self-inductance of the long, tightly wound solenoid is

The magnetic flux through N turns of wires is

The magnitude of magnetic field in the solenoid is

Solution:

Example: The length and radius of a long, tightly wound solenoid with N turns are l and R respectively. Find its self-inductance?

VnL 2

The magnetic flux through an element area dA in a rectangular region as shown in the figure is

I

I

1R

2R

r

r

l

dr

self-inductance per unite lengthIn series

1 2

Mutual Induction

1BA changing current i1 in

circuit 1 causes a changing flux 21 through circuit 2. Then an induced emf appears in circuit 2 .

The magnetic flux through circuit 2 is

12121 IMΨ

M21 is a constant that depends on the geometry of the two circuits and the magnetic properties of the material.

1IdA

emf induced in circuit 2 by changing currents in circuit 1, through mutual inductance:

emf induced in circuit 1 by changing currents in circuit 2, through mutual inductance:

Example Calculate the mutual inductance for two tightly wound concentric solenoids shown in figure below

Solution:

A current I1 in the inner solenoid sets up a magnetic field B1

the flux 21 is

The mutual inductance is

We can calculate the mutual inductance by assuming that the outer solenoid carries a current I2

The mutual inductances are equal

This is a general result

I

a

2a

23a

Solution:

rd

r

The magnetic flux through the rectangular loop is

The mutual inductance is

The mutual induced emf in the rectangular is

Example: A long, straight wire carrying a current I=I0sint and a rectangular loop whose short edges are parallel to the wire, as shown in the figure. Find the mutual inductance and the mutual induced emf in the rectangular loop.

Kirchhoff’s First Rule“Loop Rule” or “Kirchhoff’s Voltage Law (KVL)”

"When any closed circuit loop is traversed, the algebraic sum of the changes in potential must equal zero."

KVL:

Rules for potential differences across various circuit elements.

ba VV

ba VIRV

ba VC

QV

ab VV

ab VIRV

ab VC

QV

R1

R2I

IR1 IR2 0

Our convention:•We choose a direction for the current and move

around the circuit in that direction.•When a battery is traversed from the negative

terminal to the positive terminal, the voltage increases, and hence the battery voltage enters KVL with a + sign.

•When moving across a resistor, the voltage drops, and hence enters KVL with a - sign.

Modification of Kirchhoff’s loop rule:

In moving across an inductor of inductance L along the presumed direction of the current I, the potential change is = –L dI/dt

Example A device as shown below is constructed from conductors. Find the self-inductance.

Solution Suppose there is a current I in the conductor

a

a

ll

d

I

If there is a magnetic field which is parallel to the axis of the cylindrical shell, Find the current

a

a

ll

d

I

24-2 Energy in Inductors

An RL circuitWork done by the battery

Energy stored in the inductor

Energy lost in a resistor in the form of thermal energy

Loop rule

An inductor is a device for storing energy in a magnetic field.

Where is the Energy Stored?

We can integrate that term to find an expression for UL (starting from zero current)

24-3 Energy in Magnetic Fields

giving the energy density of the magnetic field:

For an ideal solenoid

We can express UL in terms of B(=nI)

Energy is stored in the magnetic field itself (just as in the capacitor / electric field case).

A general result

Energy is located within the electric and magnetic fields themselves

Example A long coaxial cable consists of two thin-walled concentric conducting cylinders with radii a and b. The inner cylinder carries a steady current I, the outer cylinder providing the return path for that current. The current sets up a magnetic field between the two cylinders. (a)  Calculate the energy stored in the magnetic field for a length l of the cable.(b)  Calculate the self inductance for a length l of the cable.

I I

I I

Solution find H by using Ampere's law

The energy density

The energy

Example An air-filled, closely wound toroidal solenoid with inner radius R1 and outer radius R2 , has

N turns. Its cross-section is rectangular. Find

magnetic-field energy.

By using Ampere’s Law, the magnitude of the magnetic field is

Solution:

The magnetic energy density

hdr r

The volume of a thin cylinder with radius r and thickness dr, as shown in the figure, is

The total magnetic energy

What is the inductance of this toroid?

24-4 Time Dependence in RL Circuits

When the switch K1 is closed, the inductor keeps the current from attaining its maximum value immediately.

Current as a function of time:

An RL circuit

L

R

1k

2k

L

II

The current in an inductor never changes instantaneously, but after the current settles down to a constant value, the inductor plays no role in the circuit

An RL circuit

L

R

1k

2k

L

II

L

R

1k

2k

IIWhen the switch is thrown to K2 the inductor keeps the current from dropping to zero immediately.

Current as a function of time: 0 t

I

RI

0

063.0 I

037.0 I

R

L

time constant

The time constant determines how fast the current changes with time

General rule: inductors resist change in current

• Hooked to current source Initially, the inductor behaves like an open switch. After a long time, the inductor behaves like an ideal wire.

• Disconnected from current source Initially, the inductor behaves like a current source. After a long time, the inductor behaves like an open switch.

Close switch Open switch

Li

demo

ACT At t=0 the switch is thrown from position b to position a in the circuit shown:

(a) I = 0 (b) I = /2R (c) I = 2/R

(a) I0 = 0 (b) I0 = /2R (c) I0 = 2/R

– What is the value of the current I0 immediately after the switch is thrown?

R

a

b

R

L

II

– What is the value of the current Ia long time after the switch is thrown?

– After a long time, the switch is thrown from position a to position b as shown:

(a) I0 = 0 (b) I0 = /2R (c) I0 = 2/R

What is the value of the current I0 just after the switch is thrown?

R

a

b

R

L

II

Just after the switch is thrown, the inductor induces an emf to keep current flowing: emf = L dI/dt (can be much larger than ) However, now there’s no place for the current to go charges build up on switch contacts high voltage across switch gapIf the electric field exceeds the “dielectric strength” (~30 kV/cm in air) breakdown SPARK!

R

a

R

L

II

If after a long time, the switch is opened abruptly from position a instead of being thrown to position b, What happens?

1) At time t = 0 the switch is closed. What is the current through the circuit immediately after the switch is closed?

a) I = 0 b) I = V/R c) I = V/2R

2) What is the current through the circuit a long time after the switch is closed?

a) I = 0 b) I = V/R c) I = V/2R

Initially, the inductor acts like an open switch

After a long time, the inductor acts like an ideal wire,

ACT

ACT At t=0, the switch is thrown from position b to position a as shown:

– Let tI be the time for circuit I to reach 1/2 of its asymptotic current.

– Let tII be the time for circuit II to reach 1/2 of its asymptotic current.

– What is the relation between tI and tII?

(a) tII < tI(b) tII = tI (c) tII > tI

a

bL

II

R

L

II

IR

a

b

L

I

R

I

We must determine the time constants of the two circuits by writing down the loop equations.

tII=2L/RtI=L/2R This confirms that inductors in series add

Time Dependence in RC Circuits

RC circuit with battery and switch

Switch at position a: battery charges capacitor

I

++ +- - -

Loop rule:

Charge as a function of time:

Current as a function of time:

Conclusion:

• Capacitor reaches its final charge(Q=C ) exponentially with time constant = RC.

• Current decays from max ( /R) with same time constant.

Suppose that the switch has been in position a for a long time, the capacity is fully charged, and there is no current. We throw the switch to position b: capacitor discharges through resistor

I

++ +- - -

Charge as a function of time:

Current as a function of time:

The negative sign indicates that the actual current is opposite in direction to the current we assumed

Conclusion:

• Capacitor discharges exponentially with time constant = RC

• Current decays from initial max value (-/R) with same time constant

Basic principle: Capacitor resists rapid change in Q resists rapid changes in V

• Charging (it takes time to put the final charge on) Initially, the capacitor behaves like a wire After a long time, the capacitor behaves like an open switch.

• Discharging Initially, the capacitor behaves like a battery. After a long time, the capacitor behaves like a wire.

ACT At t=0 the switch is thrown from position b to position a in the circuit shown: The capacitor is initially uncharged.– What is the value of the current I0+

just after the switch is thrown?

(a) I0+ = 0 (b) I0+ = /2R (c) I0+ = 2/R

a

b

R

C

II

R

(a) I = 0 (b) I = /2R (c) I > 2/R

– What is the value of the current I after a very long time?

ACT The two circuits shown below contain identical fully charged capacitors at t=0. Circuit 2 has twice as much resistance as circuit 1.

Compare the charge on the two capacitors a short time after t = 0

a) Q1 > Q2

b) Q1 = Q2

c) Q1 < Q2

Initially, the charges on the two capacitors are the same. But the two circuits have different time constants: = RC and 2 = 2RC. Since2 > 1 it takes circuit 2 longer to discharge its capacitor. Therefore, at any given time, the charge on capacitor 2 is bigger than that on capacitor 1.

24-5 Oscillations in LC Circuits

1. Set up the circuit above with capacitor, inductor, resistor, and battery.

2. Let the capacitor become fully charged.

3. Throw the switch from a to b

4. What happens?

Start with a fully charged capacitor

the loop rule:

Simple harmonic oscillator

Q0and are determined from initial conditions.

solution:

•Q0 (initial condition) determines the amplitude of the oscillations

•The frequency of the oscillations is determined by the circuit parameters (L, C)

Here,

It undergoes simple harmonic motion, just like a mass on a spring, with trade-off between charge on capacitor (Spring) and current in inductor (Mass)

ACT At t=0 the capacitor has charge Q0; the resulting oscillations have frequency 0. The maximum current in the circuit during these oscillations has value I0.

– What is the relation between 0 and 2, the frequency of oscillations when the initial charge = 2Q0?

(a) 2 = 1/2 0 (b) 2 = 0 (c) 2 = 20

(a) I = I (b) I = 2I (c) I = 4I

– What is the relation between I0 and I2, the maximum current in the circuit when the initial charge = 2Q0?

LC

+ +

- -Q Q 0

t=0

24-6 Damped Oscillations in RLC Circuits

Loop rule

Damped harmonic oscillatorsolution:

Solution:

Charge equation:

Where

And

Determines the rate of exponential damping

Q0 and are determined from initial conditions.

Comparison of an RLC circuit with and without damping

critical damping

When there is no oscillator behavior

We call this overdamping

The RLC circuit for various values of R.

critical damping

overdamping

24-7 Energy in LC and RLC Circuits

No Resistance

We take initial conditions:

Energy in capacitor

Energy in inductor

Total energy is conserved

In a pure LC circuit, energy is transferred back and forth between the capacitor’s electric field and the inductor’s magnetic field.

a resistor causes energy loss, which shows up as heat.

Resistance is Introduced

The power dissipated in the resistor

The rate of change of the energy in the capacitor and in the inductor

Equal to the power loss in the resistor

Example A 1.5 mF capacitor is charged to 57 V. The charging battery is then disconnected, and a 12 mH coil is connected in series with the capacitor so that LC oscillations occur at time t=0. (a) What is the maximum current in the coil? (b) What is the potential difference VL(t) across the inductor as a function of time?(c) What is the maximum rate (di/dt)max at which the current i changes in the circuit? Assume that the circuit contains no resistance.

Solution

(a)

Example A series RLC circuit has inductance L = 12 mH, capacitance C = 1.6 F, and resistance R = 1.5 . (a)  At what time t will the amplitude of the charge oscillations in the circuit be 50% of its initial value?(b)  How many oscillations are completed within this time?

Solution

Summary

• Definition of inductance:

• Induced emf:

• emf induced in a second loop:

• Energy in an inductor:

• Energy density of a magnetic field:

LIB

• LC circuit oscillations:

• RLC circuit oscillations:

• RL Circuits

• RC Circuits