Chapter 32: InductanceReading assignment: Chapter 32

Homework 32 (not due, use as practice for exam): OQ1, OQ3, OQ4, OQ5, QQ2, QQ3, QQ4, 1, 4, 6, 9, 16, 17, 32, 40

• Previous chapters:

• A current creates a magnetic field

• An emf and a current are induced in a wire loop when the magnetic flux through the

loop changes.

• Now:

• Self-inductance, inductance, inductor – a circuit element

• Energy in an inductor

• RL

• Mutual inductance

• RC, RLC circuits

• Distinguish between emfs and currents caused by batteries, and those induced by

changing magnetic fields.

Joseph Henry (1797- 1878)

• Final exam:

• Tuesday, Dec. 9, 9:00 am -12:00 pm (room TBA),

• Thursday, Dec. 11, 9:00 am -12:00 pm (room TBA),

• Saturday, Dec. 13, 9:00 am -12:00 pm (Olin 101)

• Comprehensive, chapters 23 – 32 (as far as we got, slide 19 (mutual inductance)

• Same format as midterms 1 and 2. Equation sheet will be provided

• Review session, Wednesday, Dec. 10, 5:00 pm – 6:00 pm

• Use homework 32 for practice for exam

Self Inductance

• Consider a solenoid L, connect it to a battery

• What happens as you close the switch?

• Lenz’s law – loop resists change in magnetic field

• Magnetic field is caused by the original current

• “Inductor” resists change in current E +–

L

dIL

dtE

A

l

Self-induced emf:

L is the inductance of coil

It is a measure of how much the inductor (coil) will resist a change in current

L is measured in Henry (1 H).

/LL

dI dt

E

L=

White board exampleInductance of a solenoid

Consider a long, uniformly wound solenoid, having N

turns, length, l, and cross-sectional area A.

(A) Find the inductance, L, of the solenoid.

(B) Calculate L for 300 turns, l = 0.25 m and cross-

sectional area A = 4 cm2.

L

dIL

dtE

20L N A

E +–

A

l

L depends on geometry and N

Inductors

• An inductor in a circuit is denoted by this symbol:

• An inductor satisfies the formula:

• L is the inductance

• Measured in Henrys (H); 1H = 1V·s/A

L

What is Kirchhoff's law for the loop above when the switch is closed?

A) E + L (dI /dt) = 0 B) E – L (dI /dt) = 0

C) None of the above D) I don’t know Kirchhoff's law for switches

• Assign currents to every path, as usual

• Kirchhoff's first law is unchanged

• The voltage change for an inductor is

• Negative if with the current

• Positive if against the current

• In steady state (dI/dt = 0), an inductor is a

wire

+

–

L

E

I

Kirchhoff's rules for Inductors

White board example

The current in a 90.0 mH inductor changes with time

as I = t2 - 6t (in SI units). Find the magnitude of the

induced emf at

(a) t = 1.00 s and

(b) t = 4.00 s.

(c) At what time is the emf zero.

RL Circuits

L

E

+ –

R

S1

Derive on board

𝐼=𝜀𝑅

(1−𝑒− 𝑡 /𝜏 )

𝜏=𝐿𝑅

An inductor slows down a change in current.

RL Circuits

L

E+–

R

𝜏=𝐿𝑅

b

White board example.

The circuit elements have the following values e = 12.0 V, R = 6.00 W, L = 30.0 mH.

A) Find the time constant of the circuit. If S1 is closed, and S2 is in position a, at what time will the current have reached 90% of its maximum value?

B) Switch S2 is at position a, and switch S1 is thrown closed at t = 0. Calculate the current in the circuit at t = 2 ms.

C) Compare the potential difference across the resistor with that across the inductor.

L

E

+ –

R

i-clicker.

When the switch is closed, the current through the circuit

exponentially approaches a value . If we repeat this experiment

with an inductor (e.g. solenoid) that has twice the number of turns

per unit length, the time it takes for the current to reach a value of

I/2

 

A increases.

B. decreases.

C. is the same.

Concept Question

L

E = 10 V

+–

The circuit at right is in a steady state. What will the voltmeter read as soon as the switch is opened?

R1 =

10

• The current remains constant at 1 A• It must pass through resistor R2

• The voltage is given by V = IR

R2 =

1 k

• Note that inductors can produce very high voltages

• Inductance causes sparks to jump when you turn a switch off

I =

1 A

1 A 1000 V IR

1000 VV

+–

Loop has unin-tended inductance

V

i-clicker.

When the switch in the circuit below is closed, the brightness of the bulb

A. Starts off at its brightest and then dims.

B. Slowly reaches its maximum brightness.

C. Immediately reaches it maximum, constant brightness.

D. Something else. + -

R

Bulb

L

(Assume the inductor

has no resistance.)

Energy in Inductors• Is the battery doing work on the inductor?

I V P+–

LEdI

ILdt

• Integral of power is work done on the inductor

U dtPdI

IL dtdt

L IdI 212 LI k

• It makes sense to say there is no energy in inductor with no current21

2U LI• Energy density inside a solenoid?

2 20

2

N AIU

0NIB

20L N A

Uu

A

2 20

22

N I

2

02

B

2

02

Bu

• Just like with electric fields, we can associate the energy with the magnetic fields, not the current carrying wires

Inductors in series and parallelL1• For inductors in series, the

inductors have the same current

• Their EMF’s add:

L2

1

dIL

dtE 2

dIL

dt 1 2

dIL L

dt

1 2L L L

• For inductors in parallel, the inductors have the same EMF but different currents

L1

L2

11

dIL

dtE

22

dIL

dtE

1 2dI dIdI

dt dt dt

1 2L L

E EL

E

1 2

1 1 1

L L L

Parallel and Series - Formulas

Capacitor Resistor Inductor

Series

Parallel

Fundamental Formula

1 2R R R

1 2

1 1 1

R R R 1 2C C C

1 2

1 1 1

C C C 1 2L L L

1 2

1 1 1

L L L

QV

C V IR

L

dIL

dtE

Mutual Inductance• Consider two solenoids sharing the same

volumeWhat happens as you close the switch?• Current flows in one coil• But Lenz’s Law wants mag. flux constant• Compensating current flows in other coil• Allows you to transfer power without

circuits being actually connected• It works even better if source is AC from

generator

E +–

I1 I2

2 1212

1

NM

I

Mutual Inductance of coil 2 w/r to coil 1 depends on flux by coil 1 through coil 2.

1 22 12 1 21

dI dIM and similarly M

dt dt

12 21M M M

1 22 1

dI dIM and M

dt dt

i-clicker.

The centers of two coils are moved closer together without

changing their relative orientation. What happens to the mutual

induction of the two coils?

A) It increases

B) It decreases

C) It stays the same

D) Not enough information.

Explain.

E +–

I1 I2

White board example.

Two solenoids A and B, spaced close to each other and sharing the same cylindrical axis, have 400 and 700 turns, respectively. A current of 3.5 A in solenoid A produces an average flux of 300 µWb through each turn of A and a flux of 90.0 µWb through each turn of B.

(a) Calculate the mutual inductance of the two solenoids.

(b) What is the inductance of solenoid A?

(c) What emf is induced in B when the current in A changes at the rate of 0.500 A/s?

E +–

IA IB

White board example.

On a printed circuit board, a relatively long straight conductor

and a conducting rectangular loop lie in the same plane, as shown

below.

If h = 0.400 mm, w = 1.30 mm, and L = 2.70 mm, what is their

mutual inductance?

i-clicker.

The primary coil of a transformer is connected to a battery, a resistor, and a switch.

The secondary coil is connected to an ammeter. When the switch is thrown closed,

the ammeter shows

 

A. zero current.

B. a nonzero current for a short instant.

C. a steady current.

D. Something else.

LC Circuits• Inductor (L) and Capacitor (C)• Let the battery charge up the capacitorNow flip the switch• Current flows from capacitor through inductor• Kirchoff’s Loop law gives:• Extra equation for capacitors:

+–

EC

L

Q

0Q C V C EI

0Q

C dI

Ldt

dQI

dt

dIQ CL

dt d dQ

CLdt dt

2

2

1d QQ

dt CL

• What function, when you take two deriva-tives, gives the same things with a minus sign?

• This problem is identical to harmonicoscillator problem

cos

sin

Q t

Q t

0 cosQ Q t

LC Circuits (2)• Substitute it in, see if it works

C

L

Q

I

0 cosQ Q t

0 sindQ

Q tdt

2

202

cosd Q

Q tdt

2

2

1d QQ

dt CL

20 0

1cos cosQ t Q t

CL

1

CL

• Let’s find the energy in the capacitor and the inductor

dQI

dt 0 sinQ t

2

2C

QU

C

2

20 cos2C

QU t

C

212LU LI 2 2 21

02 sinLQ t

2

20 sin2L

QU t

C

20 2C LU U Q C

Energy sloshes back and forth

Frequencies and Angular Frequencies• The quantity is called the angular frequency• The period is the time T you have to wait for it to repeat• The frequency f is how many times per second it repeats

2T

1

CL 0 cosQ Q t T

1f T

2 f

WFDD broadcasts at 88.5 FM, that is, at a frequency of 88.5 MHz. If they generate this with an inductor with L = 1.00 H,

what capacitance should they use?

2 f 6 12 88.5 10 s 8 15.56 10 s 2 1LC

2

1C

L

28 1 6

1

5.56 10 s 10 H

3.23 pFC

White board example.

A 1.00 µF capacitor is charged by a 40 V power supply. The fully-

charged capacitor is then discharged through a 10.0 mH inductor.

Find the maximum current in the resulting oscillations.

RLC Circuits• Resistor (R), Inductor (L), and Capacitor (C)• Let the battery charge up the capacitorNow flip the switch• Current flows from capacitor through inductor• Kirchoff’s Loop law gives:• Extra equation for capacitors:

+–

EC

L

Q

I

0Q dI

L RIC dt

dQI

dt 2

20

Q dQ d QR L

C dt dt

• This equation is hard to solve, but not impossible• It is identical to damped, harmonic oscillator

20 cosRt LQ Q e t

R

2

2

1

4

R

LC L