hittTwo long straight wires pierce the plane of the paper at vertices of an equilateral triangle as shown. They each carry 3A in the same direction. The magnetic field at the third vertex (P) has the magnitude and direction (North is up):

(1) 20 μT, west (2) 17 μT, east (3) 15 μT, north (4) 26 μT, south (5) none of these

4 cm

P

4 cm

Long, straight, parallel wires carry equal currents into or out of page. Rank according to the magnitude of the force on the central wire, largest to smallest.

1. d, c, a, b 2. a, b, c, d 3. b, c, d, a 4. c, a, b, d5. b, d, c, a

A horizontal power line carries a current of 5000 A from south to north. Earth's magnetic eld (60 μT) is directed towards north and inclined down-ward 50 degrees to the horizontal. Find the magnitude and direction of the magnetic force on 100 m of the line due to the Earth's field.

(1)23 N, west (2) 23 N, east (3) 30N, west

(4) 30N, east (5) none of these

50

B

N

I

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)

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 acurately predicted by what is

known as Lenz's

Lenz's Rule

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 haces the corresponding north pole of the b

B

ar magnet

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

the change in which generated the induced current.repels

(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 :

B

As the bar magnet approaches the loop the magnet field points towards the left

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

of the loop magnetic flux also

B

increases. The induced current flows in the

(CCW) direction so that the induced magnetic field opposes

the magnet field . The net field . The induced current is t

i

net i

B

B B B B

counterclockwise

B B

hus trying

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

the induced current in the first place.

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 :

B

As the bar magnet moves away from the loop the magnet field points towards the left

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

of the loop magnetic flux

B

also decreases. The induced current flows in the

(CW) direction so that the induced magnetic field adds to

the magnet field . The net field . The induced current is thus

i

net i

B

B B B B

clockwise

B B

trying

to from decreasing. Remember it was the decrease in that generated

the induced current in the first place.

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 :

B

As the bar magnet approaches the loop the magnet field points towards the right

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

of the loop magnetic flux als

B

o increases. The induced current flows in the

(CW) direction so that the induced magnetic field opposes

the magnet field . The net field . The induced current is thus try

i

net i

B

B B B B

clockwise

B B

ing

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

the induced current in the first place.

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 magnet field points towards the

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

the magnitude of the loop magnetic flux

B

B also decreases. The induced current

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

adds to the magnet field . The net field . The induced cur

i

net i

B

B B B B

counterclockwise

B B

rent is thus

trying to from decreasing. Remember it was the decrease in that

generated the induced current in the first place.

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-magnetic fie

Induction and energy transfers

do work

ld 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

Consid

R

er 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 constant

sp

L

B

eed . The magnetic flux through the loop

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

2 3

The rate at which thermal energy is dissipated on

( )

The magnetic forces on the wire sides are shown

in the figure. Forces and cancel each other.

Force

th

R

BLv B L vP i R R

R R

F F

F

eqs.1

1 1

2 2

1

2 2 2

1

sin 90

The rate at which the external agent is

producing mechanical work ( )

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

me

ext

BLviL B F iLB iLB LB

R

B L vF

R

B L vP F v

R

eqs.2

chanical 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 mechanical energy that moves the plate is

transformed into thermal energy that heats up the pla

eddy currents

te.

(30 –13)

Consider the copper ring of radius shown in the

figure. It is paced in a uniform magnetic field

pointing into the page, that icreases as function

of time. The resulting chan

r

B

Induced electric fields

ge in magnetic flux

induces a current i in the counterclock-wise

(CCW) direction.

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

electric field must be present in order to set the electrons on 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 absense 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

in 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 eqs.1 with eqs.2

we get:

B

E ds

d

dtd

E ds

eqs.1

eqs.2

E

E

2 2

2

cos 0 2

22

B

BB

E ds Eds E ds rE

d dBr B r

dt dtdB

rE rd

dt

r dBE

dtt

(30 –15)

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 unifrom

magnetic field ino

N

NA n

i

B ni

Inductance

side the solenoid.

The solenoid magnetic flux B NBA

B

2The 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 solenoid

B o

B

N n n A i

Li L

inductance

22

. The inductance depends

on the geometry of the particular inductor.

For the solenoid oBo

n AiL n A

i i

Inductance of the solenoid

2 oL 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

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 determined the re

B

B

Li

d diL

dt dt

the Henry (symbol: H)

An inductor has inductance

sulting emf known as

em

1 H if a current

change of 1 A/s results in a self-induced emf of

f

1 V

.

.

B

L

d diL

dt dt

SI unit for L :

Eself induced

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 a and the equivelent circuit is shown in

the lower figure. It cont

t

RL circuits

ains 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 function of time . We

write Kirchhoff's

R L

RL

i t

E

loop rule starting at point x and

moving around the loop in the clockwise direction.

0 di

L iRd

dt

iiR L

dt 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 th( 1 e) "t Li t e

R

i

R

E

time cons " of

the RL circuit.

tant

/( ) 1 ti t eR

E L

R

/

/

/

( ) 1 Here

The voltage across the resistor 1 .

The voltage across the inductor

The solution gives 0 at 0 as required by the

initial conditio

t

tR

tL

Li t e

R R

V iR e

diV L e

dti t

E

E

E

n. 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 c

E

E

E

E

the current, for all

practical purposes has reached its terminal value / . R

onstants

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. Consider t

Energy stored in a magnetic field

he circuit

shown in the figure. Kirchhoff's loop rules gives:

2

2

If we multiply both sides of the equation we get:

The term describes the rate at which the batter delivers energy to the circuit

The term is the rate at which

di diL iR i Li i R

dt dti

i R

E E

E

thermal 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

diLi

dtdU di

Li dU Lididt 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 soleniod of length and loop area

that has windings per unit length. The solenoid

carries a current i that generates a uniform magnetic

field o

A

n

B ni

Energy density of a magnetic field

inside the solenoid. The magnetic field

outside the solenoid is approximately zero.

2 221

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 in

oB

BB

n A iU Li

Uu V

V

22 2 2 2 2 2 2

side

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.

o o oB

o o

n A i n i n i Bu

A

2

2B

o

Bu

(30 –21)

N2

N1 Consider two inductors

which are placed close enough

so that the magnetic field of one

can influence the other.

Mutual Induction

1 1

2 21 1

1

In fig.a 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 curent varies wit

i B

M i

i

2 12 21 21

h 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 their relative position. (30 –22)

12 21

diM

dtE

N2

N1

2 2

1 12 2

2

In fig.b 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 curent varies wit

i B

M i

i

1 21 12 12

h 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 we

d diM M

dt dt

E

ll as their relative position.

21 12

diM

dtE

(30 –23)

N2

N1

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 geometr

M M M M M

M

mutual inductance

the He

y of th

nry (H)

e two

inductors as well as their relative position.

The expressions for the induced emfs across the two inductors become:

The SI unit for M :

21

diM

dtE 1

2

diM

dtE

(30 –24)

1 2Mi 2 1Mi

Electrons are going around a circle in a counterclockwise direction as shown. At the center of the circle they produce a magnetic field that is:

A. into the pageB. out of the pageC. to the leftD. to the rightE. zero

e

Long parallel wires carry equal currents into or out of the page. Rank according to the magnitude of the net magnetic field at the center of the square.

1. C,D, (A,B) 2. A, B, (C,D) 3. B, A, C, D4. D, (A, B), C

Partial Loops (cont.)

• Note on problems when you have to evaluate a B field at a point from several partial loops

– Only loop parts contribute, proportional to angle (previous slide)

– Straight sections aimed at point contribute exactly nothing

– Be careful about signs, e.g.in (b) fields partially cancel, whereas in (a) and (c) they add

In a series of experiments Michael Faraday in England

and Joseph Henry in the US were able to generate

electric currents without the use of batteries

Below we describe some of the

Faraday's experiments

se experiments that

helped formulate whats 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 we

1.

2.

3. 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 appears induced current

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 as

" "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

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

all of which are statements of experim

Maxwell's equations

ental 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:

BMagnetic Flux Φ

1 we divide the surface that has the loop as its border

into area 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 a

Bd BdA

n B

2.

3.

2: T m known as the Weber (symbol

ll the terms. cos

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

W

g

b)

form:

B BdA B dA

SI magnetic flux unit

B

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

at which the magnetic flux Φ through the loop changes with time

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 angle

BBMethods for changing Φ through a loop

1.

2.

3.

ˆ between and

Problem 30-11

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)