Chapter 30 Induction and Inductance

In this chapter we will study the following topics:

-Faraday’s law of induction -Lenz’s rule -Electric field induced by a changing magnetic field -Inductance and mutual inductance - RL circuits -Energy stored in a magnetic field

(30–1)

In a series of experiments, Michael Faraday in England

and Joseph Henry in the U.S. were able to generate

electric currents without the use of batteries.

Below

Faraday's

3

0.2 Two exp

Experiment

e

s

riments

we describe some of these experiments that

helped formulate what is known as "Faraday's law

of induction."The circuit shown in the figure consists of a wire loop connected to a sensitive

ammeter (known as a "galvanometer"). If we approach the loop with a permanent

magnet we see a current being registered by the galvanometer. The results can be

summarized as follows:

A current appears only if there is relative motion between the magnet and the loop.

Faster motion results in a larger current.

If

1.

2.

3. we reverse the direction of motion or the polarity of the magnet, the current

reverses sign and flows in the opposite direction.

The current generated is known as " "; the emf that appearinduced current s

is known as " "; the whole effect is called " "induced emf induction. (30–2)

In the figure we show a second type of experiment

in which current is induced in loop 2 when the

switch S in loop 1 is either closed or opened. When

the current in loop 1 is constant no induced current

is observed in loop 2. The conclusion is that the

magnetic field in an induction experiment can be

generated either by a permanent magnet or by an

electric current in a coil.

loop 1loop 2

Faraday summarized the results of his experiments in what is known a

30-3 Faraday's Law of Induction

s

" "Faraday's law of induction.An emf is induced in a loop when the number of magnetic field lines that

pass through the loop is changing.

Faraday's law is not an explanation of induction but merely a description of

what induction is. It is one of the four " of electromagnetism,"

all of which are statements of experimen

Maxwell's equations

tal results. We have already encountered

Gauss' law for the electric field, and Ampere's law (in its incomplete form).(30–3)

B

n̂

dA

The magnetic flux through a surface that borders

a loop is determined as follows:

Magnetic Flux Φ

1

B

We divide the surface that has the loop as its border

into elements of area . dA

.

For each element we calculate the magnetic flux through it: cos .

ˆHere is the angle between the normal and the magnetic field vectors

at the position of the element.

We integrate

Bd BdA

n B

2.

3.

2: T m known as the Weber (symbol

all the terms. cos

We can express Faraday's law of induction in the followin

Wb).

g form:

B BdA B dA

SI magnetic flux unit

The magnitude of the emf induced in a conductive loop is equal to the

rate at which the magnetic flux Φ through the loop changes with time.B

E

B B dA

Bd

dt

E

(30–4)

B

n̂

dA

cosB BdA B dA

Change the magnitude of within the loop.

Change either the total area of the coil or

the portion of the area within the magnetic field.

Change the angl

B

Methods for Changing Φ Through a Loop

1.

2.

3.

B

ˆe between and .

Problem 30-13

cos cos

sin

2

2 sin 2

B

B

B n

NAB NabB t

dNabB t

dtf

fNabB t

An Example.

E

E

B

n̂

loop(30–5)

Bd

dt

E

We now concentrate on the negative sign

in the equation that expresses Faraday's law.

The direction of the flow of induced current

in a loop is accurately predicted by what is

known as L

30 - 4 Lenz's Rule

enz's rule.

An induced current has a direction such that the magnetic field due to the

induced current opposes the change in the magnetic flux that induces the current.

Lenz's rule can be implemented using one of two methods:

In the figure we show a bar magnet approaching a loop. The induced current flows

in the direction indicated becaus

1. Opposition to pole movement

e this current generates an induced magnetic field

that has the field lines pointing from left to right. The loop is equivalent to a

magnet whose north pole faces the corresponding north pole of the bar magnet

approaching the loop. The loop the approaching magnet and thus opposes

the change in that generated the induced current.Brepels

(30–6)

N S

magnet motion

Bar magnet approaches the loop

with the north pole facing the loop.

2. Opposition to flux change

Example :a

As the bar magnet approaches the loop, the magnetic field points toward the left

and its magnitude increases with time at the location of the loop. Thus the magnitude

of the loop magnetic flux alB

B

net

so increases. The induced current flows in the

(CCW) direction so that the induced magnetic field opposes

the magnetic field . The net field . The induced current

i

i

B

B B B B

counterclockwise

is thus trying

to from increasing. Remember that it was the increase in that

generated the induced current in the first place. B B prevent

(30–7)

N S

magnet motion

Bar magnet moves away from the loop

with north pole facing the loop.

2. Opposition to flux change

Example :b

As the bar magnet moves away from the loop, the magnetic field points

toward the left and its magnitude decreases with time at the location of the

loop. Thus the magnitude of the loop magnetic flu

B

net

x also decreases. The

induced current flows in the (CW) direction so that the induced

magnetic field adds to the magnetic field . The net field .

The induced current

B

i iB B B B B

clockwise

is thus trying to from decreasing. Remember

that it was the decrease in that generated the induced current in the first

place.

B

B

prevent

(30–8)

S N

magnet motion

Bar magnet approaches the loop

with south pole facing the loop.

2. Opposition to flux change

Example :c

As the bar magnet approaches the loop, the magnetic field points toward the

right and its magnitude increases with time at the location of the loop. Thus

the magnitude of the loop magnetic flux B

B

net

also increases. The induced current

flows in the (CW) direction so that the induced magnetic field

opposes the magnetic field . The net field . The induced current is

t

i

i

B

B B B B

clockwise

hus trying to from increasing. Remember that it was the increase in

that generated the induced current in the first place. B

B

prevent

(30–9)

S N

magnet motion

Bar magnet moves away from the loop

with south pole facing the loop.

2. Opposition to flux change

Example :d

As the bar magnet moves away from the loop, the magnetic field points toward

the right and its magnitude decreases with time at the location of the loop. Thus

the magnitude of the loop magnetic fl

B

net

ux also decreases. The induced current

flows in the (CCW) direction so that the induced magnetic

field adds to the magnetic field . The net field . The induced

B

i iB B B B B

counterclockwise

B

current is thus trying to from decreasing. Remember that it was the

decrease in that generated the induced current in the first place. B

prevent

(30–10)

By Lenz's rule, the induced current always opposes

the external agent that produced the induced current.

Thus the external agent must always on the

loop-magneti

30 - 5 Induction and Energy Transfers

do work

c field system. This work appears as

thermal energy that gets dissipated on the resistance

of the loop wire.

Lenz's rule is actually a different formulation of

the principle of energy conservation.

R

Consider the loop of width shown in the figure.

Part of the loop is located in a region where a

uniform magnetic field exists. The loop is being

pulled outside the magnetic field region with cons

L

B

tant

speed . The magnetic flux through the loop is

. The flux decreases with time:

B

B

v

BA BLx

d dx BLvBL BLv i

dt dt R R

EE

(30–11)

2 2 2 22

th

2 3

The rate at which thermal energy is dissipated on is

( )

The magnetic forces on the wire sides are shown

in the figure. Forces and cancel each other:

Force

R

BLv B L vP i R R

R R

F F

eq. 1

1 1

2 2

1

2 2 2

ext 1

, sin 90 ,

. The rate at which the external agent is

producing mechanical work is ( )

If we compare equations 1 and 2 we see that indeed the

BLvF iL B F iLB iLB LB

R

B L vF

R

B L vP Fv

R

eq. 2

mechanical work done by the external agent that moves

the loop is converted into thermal energy that appears

on the loop wires.

(30–12)

We replace the wire loop in the previous example

with a solid conducting plate and move the plate

out of the magnetic field as shown in the figure.

The motion between the plate and indB

Eddy Currents

uces a

current in the conductor and we encounter an opposing

force. With the plate, the free electrons do not follow

one path as in the case of the loop. Instead, the electrons

swirl around the plate. These currents are known as

" " As in the case of the wire loop, the net

result is that the mechanical energy that moves the plate is

transformed into thermal energy that heats up

eddy currents.

the plate.

(30–13)

Consider the copper ring of radius shown in the

figure. It is placed in a uniform magnetic field

pointing into the page, which increases as a function

of time. The resu

r

B

30 - 6 Induced Electric Fields

lting change in magnetic flux

induces a current in the counterclockwise

(CCW) direction.

i

The presence of the current in the conducting ring implies that an induced

electric field must be present in order to set the electrons in motion.

Using the argument above we can reformulate Farada

i

E

y's law as follows:

A changing magnetic field produces an electric field.

The induced electric field is generated even in the absence of the

copper ring.

Note :

(30–14)

Consider the circular closed path of radius shown in

the figure to the left. The picture is the same as that

on the previous page except that the copper ring has

been removed. The path is now an a

r

bstract line.

The emf along the path is given by the equation

( )

The emf is also given by Faraday's law:

( ). If we compare eq. 1 with eq. 2

we get:

B

E d

dE d

s

d

dt

s

eq. 1

eq. 2

E

E

2 2

2

cos 0

22

.

2

BB

B

E ds Eds E ds rE

d dBr B r

dt dtdB

rE rd

dt

r dBE

dtt

(30–15)

0

Consider a solenoid of length that has loops of

area each, and windings per unit length. A current

flows through the solenoid and generates a uniform

magnetic field

N

NA n

i

B

30 - 7 Inductance

inside the solenoid.

The solenoid magnetic flux is .B

ni

NBA

B

20The total number of turns . The result we got for the

special case of the solenoid is true for any inductor: . Here is a

constant known as the of the solenoi

B

B

N n n A i

Li L

inductance

220

0

d. The inductance depends

on the geometry of the particular inductor.

For the solenoid, .B n AiL n A

i i

Inductance of the Solenoid

20 L n A

(30–16)

loop 1loop 2In the picture to the right we

already have seen how a change

in the current of loop 1 results

in a change in the flux through

loop 2, and thus creates an

induced emf in loop 2.

30 - 8 Self - Induction

If we change the current through an inductor this causes

a change in the magnetic flux through the inductor

according to the equation . Using Faraday's

law we can determine the resu

B

B

Li

d diL

dt dt

the henry (symbol: H)

An inductor has inductance 1 H if a current

change of 1 A/s results in a self

lting emf known as

-induced emf o

f

emf:

1 V

.

.

Bd diL

dt dt

L

SI unit for :

self - induc

L

ed E

di

Ldt

E

(30–17)

Consider the circuit in the upper figure with the switch

S in the middle position. At 0 the switch is thrown

in position and the equivalent circuit is shown in

the lower figure.

t

a

30 - 9 RL Circuits

It contains a battery with emf ,

connected in series to a resistor and an inductor

(thus the name " circuit"). Our objective is to

calculate the current as a function of time . We

write Kir

R L

RL

i t

E

chhoff's loop rule starting at point and

moving around the loop in the clockwise direction:

0 di

L iR

x

diiR L

t dd t E E

/

The initial condition for this problem is (0) 0. The solution of the differential

equation that satisfies the initial condition is

The constant is known as the "( ) 1 .t Li t e

R R

i

time constantE

" of

the circuit. RL

/( ) 1 ti t eR

E L

R

(30–18)

/

/

/

( ) 1 Here .

The voltage across the resistor 1 .

The voltage across the inductor .

The solution gives 0 at 0 as required by the

initial conditi

t

tR

tL

Li t e

R R

V iR e

diV L e

dti t

E

E

E

on. The solution gives ( ) / .

The circuit time constant / tells us how fast

the current approaches its terminal value:

( ) 0.632 /

( 3 ) 0.950 /

( 5 ) 0.993 /

If we wait only

i R

L R

i t R

i t R

i t R

a few time

E

E

E

E

the current, for all

practical purposes, has reached its terminal value / . R

constants

E .

(30–19)

We have seen that energy can be stored in the electric field

of a capacitor. In a similar fashion, energy can be stored in

the magnetic field of an inductor. Co

30 - 10 Energy Stored in a Magnetic Field

nsider the circuit

shown in the figure. Kirchhoff's loop rule gives

2

2

. If we multiply both sides of the equation we get: .

The term describes the rate at which the battery delivers energy to the circuit.

The term is the rate at which t

di diL iR i Li i Rdt dt

i

i R

E E

E

hermal energy is produced on the resistor.

Using energy conservation we conclude that the term is the rate at which

energy is stored in the inductor: . We integrate

both

BB

diLidt

dU diLi dU Lidi

dt dt

2 2

sides of this equation: .2 2

ii

oB

o

L i LiU Li di

2

2B

LiU

(30–20)

B Consider the solenoid of length and loop area

that has windings per unit length. The solenoid

carries a current that generates a uniform magnetic

field

A

n

i

30 - 11 Energy Density of a Magnetic Field

0 inside the solenoid. The magnetic field

outside the solenoid is approximately zero.

B ni

2 22 01

The energy stored by the inductor is equal to .2 2

This energy is stored in the empty space where the magnetic field is present.

We define as energy density where is the volume

B

BB

n A iU Li

Uu V

V

2 2 2 2 2 2 2 20 0 0

0 0

inside

the solenoid. The density .2 2 2 2

This result, even though it was derived for the special case of a uniform

magnetic field, holds true in general.

B

n A i n i n i Bu

A

2

0

2B

Bu

(30–21)

N2N1 Consider two inductors

that are placed close enough

so that the magnetic field of one

can influence the other.

30 - 12 Mutual Induction

1 1

2 21 1

1

In fig. we have a current in inductor 1. That creates a magnetic field

in the vicinity of inductor 2. As a result, we have a magnetic flux

through inductor 2. If current varies w

a i B

M i

i

2 12 21 21

ith time, then we have a time-varying

flux through inductor 2 and therefore an induced emf across it:

. is a constant that depends on the geometry

of the two inductors as we

d diM M

dt dt

E

ll as on their relative position. (30–22)

12 21

diM

dtE

N2N1

2 2

1 12 2

2

In fig. we have a current in inductor 2. That creates a magnetic field

in the vicinity of inductor 1. As a result, we have a magnetic flux

through inductor 1. If current varies w

b i B

M i

i

1 21 12 12

ith time, then we have a time-varying

flux through inductor 1 and therefore an induced emf across it:

. is a constant that depends on the geometry

of the two inductors as w

d diM M

dt dt

E

ell as on their relative position. 21 12

diM

dtE

(30–23)

N2N1

12 21

diM

dtE

21 12

diM

dtE

12 21 12 21It can be shown that the constants and are equal: .

The constant is known as the " " between the two coils.

Mutual inductance is a constant that depends on the geomet

M M M M M

M

mutual inductance

th

ry

e h

of the

enry (

two

inductors as well as on their relative position.

The expressions for the induced emfs across the two inductors be

)

ome:

H

c

The SI unit for :M

21

diMdt

E 12

diMdt

E

(30–24)

1 2Mi 2 1Mi