PH 222-2C Fall 2012

Induction and inductanceLecture 15

Chapter 30(Halliday/Resnick/Walker, Fundamentals of Physics 8th edition)

1

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

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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 we describe some of

Faraday's Experiments

these experiments thathelped formulate what is known as "Faraday's lawof induction."

The circuit shown in the figure consists of a wire loop connected to a sensitiveammeter (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 currentreverses 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. 3

In the figure we show a second type of experimentin which current is induced in loop 2 when the switch S in loop 1 is either closed or opened. Whenthe current in loop 1 is constant no induced currentis observed in loop 2. The conclusion is that the magnetic field in an induction experiment can begenerated either by a permanent magnet or by anelectric 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 thatpass 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 equationstal results. We have already encountered

Gauss' law for the electric field, and Ampere's law (in its incomplete form). 4

B

n̂

dA

The magnetic flux through a surface that bordersa loop is determined as follows:

Magnetic Flux Φ

1

B

We divide the surface that has the loop as its borderinto 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 followinWb).

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

B B dA

Bddt

5

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

BMethods for Changing Φ Through a Loop1.2.

3.

B

ˆe between and .

Problem 30-11cos cos

sin

22 sin 2

B

B

B n

NAB NabB td NabB tdtffNabB ft

An Example.

B

n̂

loop6

Bddt

We now concentrate on the negative sign in the equation that expresses Faraday's law.The direction of the flow of induced currentin a loop is accurately 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 flowsin the direction indicated becaus

1. Opposition to pole movement

e this current generates an induced magnetic fieldthat 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 magnetapproaching the loop. The loop the approaching magnet and thus opposes the change in that generated the induced current.B

repels

7

N S

magnet motion

Bar magnet approaches the loop with the north pole facing the loop.

2. Opposition to flux changeExample :a

As the bar magnet approaches the loop, the magnetic field points toward the leftand its magnitude increases with time at the location of the loop. Thus the magnitudeof 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

8

N S

magnet motion

Bar magnet moves away from the loop with north pole facing the loop.

2. Opposition to flux changeExample :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

9

S N

magnet motion

Bar magnet approaches the loop with south pole facing the loop.

2. Opposition to flux changeExample :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

10

S N

magnet motion

Bar magnet moves away from the loop with south pole facing the loop.

2. Opposition to flux changeExample :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 currentflows 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

Bcurrent is thus trying to from decreasing. Remember that it was the decrease in that generated the induced current in the first place. B

prevent

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By Lenz's rule, the induced current always opposesthe external agent that produced the induced current.Thus the external agent must always on theloop-magnetic fie

Induction and Energy Transfers

do workld 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 ofthe principle of energy conservation.Consi

R

der 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 beingpulled outside the magnetic field region with constants

L

B

peed . The magnetic flux through the loop is. The flux decreases with time:

B

B

vBA BLx

d dx BLvBL BLv idt dt R R

12

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 RR 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 LBR

B L vFR

B L vP FvR

eq. 2

mechanical work done by the external agent that movesthe loop is converted into thermal energy that appearson the loop wires.

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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 followone path as in the case of the loop. Instead, the electronsswirl around the plate. These currents are known as " " As in the case of the wire loop, the netresult is that the mechanical energy that moves the plate is transformed into thermal energy that heats up

eddy currents.

the plate.

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Consider the copper ring of radius shown in thefigure. It is placed in a uniform magnetic field pointing into the page, which increases as a function of time. The resulting

rB

Induced Electric Fields

change in magnetic fluxinduces 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 :

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Consider the circular closed path of radius shown in the figure to the left. The picture is the same as thaton 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

ddt

s

eq. 1

eq. 2

2 2

2

cos 0

22

.

2

BB

B

E ds Eds E ds rE

d dBr B rdt dt

dBrE rd

dt

r dBEdtt

16

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 in

NNA n

iB ni

Inductance

side the solenoid.The solenoid magnetic flux is .B 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 Ai i

Inductance of the Solenoid

20 L n A

17

loop 1loop 2In the picture to the right we

already have seen how a changein 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 causesa change in the magnetic flux through the inductor

according to the equation . Using Faraday's

law we can determine the resu

B

B

Lid diLdt 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 diLdt dt

L

SI unit for :

self - induc

L

ed

diLdt

18

Consider the circuit in the upper figure with the switchS in the middle position. At 0 the switch is thrownin position and the equivalent circuit is shown inthe lower figure. It cont

ta

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 a function of time . We write Kirchhoff'

R LRL

i t

s loop rule starting at point and moving around the loop in the clockwise direction:

0 diL iR

x

diiR Lt dd t

/

The initial condition for this problem is (0) 0. The solution of the differentialequation that satisfies the initial condition is

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

i

time constant " of

the circuit. RL

/( ) 1 ti t eR

L

R

19

/

/

/

( ) 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 eR R

V iR e

diV L edt

i t

on. The solution gives ( ) / .The circuit time constant / tells us how fastthe current approaches its terminal value: ( ) 0.632 /

( 3 ) 0.950 /

( 5 ) 0.993 /If we wait only

i RL R

i t R

i t R

i t R

a few time

the current, for all

practical purposes, has reached its terminal value / . R constants

20

We have seen that energy can be stored in the electric fieldof a capacitor. In a similar fashion, energy can be stored inthe magnetic field of an inductor. Consider

Energy Stored in a Magnetic Field

the circuitshown 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

ii R

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 Lididt dt

2 2

sides of this equation: .2 2

ii

oB

o

L i LiU Li di

2

2B

LiU

21

B

0

Consider the solenoid of length and loop area that has windings per unit length. The solenoidcarries a current that generates a uniform magnetic field

An

iB ni

Energy Density of a Magnetic Field

inside the solenoid. The magnetic fieldoutside the solenoid is approximately zero.

2 22 01The energy stored by the inductor is equal to .

2 2This 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 VV

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 uniformmagnetic field, holds true in general.

Bn A i n i n i Bu

A

2

0

2BBu

22

N2N1 Consider two inductors

that are placed close enough so that the magnetic field of onecan influence the other.

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 BM 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 Mdt dt

ll as on their relative position.

12 21

diMdt

23

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 BM 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 Mdt dt

ell as on their relative position.

21 12

diMdt

24

N2N1

12 21

diMdt

21 12

diMdt

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 MM

mutual inductance

thry

e hof the

enry ( two

inductors as well as on their relative position. The expressions for the induced emfs across the two inductors be

)ome:

Hc

The SI unit for :M

21 diM

dt 1

2diMdt

1 2Mi 2 1Mi 25