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Chapter 32

Inductance

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Some Terminology Use emfand currentwhen they are

caused by batteries or other sources

Use induced emfand induced currentwhen they are caused by changingmagnetic fields

When dealing with problems inelectromagnetism, it is important todistinguish between the two situations

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Self-Inductance When the switch is

closed, the current

does notimmediately reach

its maximum value

Faradays law can

be used to describethe effect

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Self-Inductance, 2 As the current increases with time, the

magnetic flux through the circuit loop

due to this current also increases withtime

This corresponding flux due to thiscurrent also increases

This increasing flux creates an inducedemf in the circuit

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Self-Inductance, 3 The direction of the induced emf is such that

it would cause an induced current in the loop

which would establish a magnetic fieldopposing the change in the original magnetic

field

The direction of the induced emf is opposite

the direction of the emf of the battery

This results in a gradual increase in the

current to its final equilibrium value

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Self-Inductance, 4 This effect is called self-inductance

Because the changing flux through the

circuit and the resultant induced emf arisefrom the circuit itself

The emfL is called a self-induced

emf

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Self-Inductance, Coil Example

A current in the coil produces a magnetic field

directed toward the left (a) If the current increases, the increasing flux creates an

induced emf of the polarity shown (b) The polarity of the induced emf reverses if the current

decreases (c)

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Self-Inductance, Equations A induced emf is always proportional to

the time rate of change of the current

L is a constant of proportionality called

the inductance of the coil and itdepends on the geometry of the coiland other physical characteristics

IL

d Ldt

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Inductance of a Coil A closely spaced coil ofNturns carrying

currentIhas an inductance of

The inductance is a measure of the

opposition to a change in current

I I

B LNLd dt

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Inductance Units The SI unit of

inductance is the

henry (H)

Named for JosephHenry (pictured

here)

A

sV1H1

=

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Inductance of a Solenoid Assume a uniformly wound solenoid

having Nturns and length Assume is much greater than the radius

of the solenoid

The interior magnetic field is

I Io oNB n l

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Inductance of a Solenoid, cont The magnetic flux through each turn is

Therefore, the inductance is

This shows that L depends on thegeometry of the object

IB o

NA

BA l

2

I

B oN N A

L

l

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RL Circuit, Introduction A circuit element that has a large self-

inductance is called an inductor

The circuit symbol is We assume the self-inductance of the

rest of the circuit is negligible compared

to the inductor However, even without a coil, a circuit will

have some self-inductance

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Effect of an Inductor in a

Circuit The inductance results in a back emf

Therefore, the inductor in a circuit

opposes changes in current in that

circuit

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RL Circuit, Analysis An RL circuit contains an

inductor and a resistor

When the switch is closed(at time t= 0), the current

begins to increase

At the same time, a back

emf is induced in theinductor that opposes the

original increasing current

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Active Figure 32.3

(SLIDESHOW MODE ONLY)

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RL Circuit, Analysis, cont. Applying Kirchhoffs loop rule to the

previous circuit gives

Looking at the current, we find

0I

Id

R Ldt

1I Rt L

eR

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RL Circuit, Analysis, Final The inductor affects the current

exponentially

The current does not instantly increaseto its final equilibrium value

If there is no inductor, the exponential

term goes to zero and the current wouldinstantaneously reach its maximumvalue as expected

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RL Circuit, Time Constant The expression for the current can also be

expressed in terms of the time constant, , of

the circuit

where = L / R

Physically, is the time required for the

current to reach 63.2% of its maximum value

1I t

eR

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RL Circuit, Current-Time

Graph, (1) The equilibrium value

of the current is /R

and is reached as tapproaches infinity

The current initiallyincreases veryrapidly

The current thengradually approaches

the equilibrium value

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RL Circuit, Current-Time

Graph, (2) The time rate of

change of the

current is amaximum at t= 0

It falls off

exponentially as t

approaches infinity In general,

I td edt L

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Active Figure 32.6


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Energy in a Magnetic Field In a circuit with an inductor, the battery

must supply more energy than in a

circuit without an inductor Part of the energy supplied by the

battery appears as internal energy inthe resistor

The remaining energy is stored in themagnetic field of the inductor

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Energy in a Magnetic Field,

cont. Looking at this energy (in terms of rate)

Iis the rate at which energy is beingsupplied by the battery

I2Ris the rate at which the energy is beingdelivered to the resistor

Therefore, LIdI/dtmust be the rate at whichthe energy is being stored in the magnetic

field

2 I I I I

d R L

dt

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Energy in a Magnetic Field,

final Let Udenote the energy stored in the

inductor at any time

The rate at which the energy is stored is

To find the total energy, integrate and

II

dU dL

dt dt

0

I

I IU L d

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Energy Density of a Magnetic

Field Given U = LI2

SinceAis the volume of the solenoid, themagnetic energy density, uB is

This applies to any region in which a magnetic

field exists (not just the solenoid)

22

21

2 2o

o o

B B

U n A A n

l l

2

2B

o

U Bu

A

l

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Example: The Coaxial Cable Calculate L for the cable

The total flux is

Therefore, L is

The total energy is

ln2 2

I Ib o oB

a bB dA dr

r a

ll

ln2I

B o bL a

l

2

21ln

2 4

I

Io b

U L a

l

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Mutual Inductance, 3 The mutual inductanceM12 of coil 2

with respect to coil 1 is

Mutual inductance depends on thegeometry of both circuits and on their

orientation with respect to each other

2 12

12

1I

NM

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Induced emf in Mutual

Inductance If currentI1 varies with time, the emf

induced by coil 1 in coil 2 is

If the current is in coil 2, there is a

mutual inductance M21 If current 2 varies with time, the emf

induced by coil 2 in coil 1 is

12 12 2 12

Id d N Mdt dt

2

1 21

Id M

dt

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Mutual Inductance, Final In mutual induction, the emf induced in one

coil is always proportional to the rate at which

the current in the other coil is changing The mutual inductance in one coil is equal to

the mutual inductance in the other coil M12 = M21 = M

The induced emfs can be expressed as

2 1

1 2and

I Id d M M

dt dt

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LCCircuits A capacitor is

connected to aninductor in an LCcircuit

Assume the capacitoris initially charged andthen the switch isclosed

Assume no resistanceand no energy lossesto radiation

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Oscillations in an LCCircuit Under the previous conditions, the current in

the circuit and the charge on the capacitoroscillate between maximum positive andnegative values

With zero resistance, no energy istransformed into internal energy

Ideally, the oscillations in the circuit persistindefinitely The idealizations are no resistance and no

radiation

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Oscillations in an LCCircuit, 2 The capacitor is fully charged

The energy Uin the circuit is stored in the

electric field of the capacitor The energy is equal to Q2max / 2C

The current in the circuit is zero

No energy is stored in the inductor The switch is closed

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Oscillations in an LCCircuit, 3 The current is equal to the rate at which

the charge changes on the capacitor

As the capacitor discharges, the energystored in the electric field decreases

Since there is now a current, some energyis stored in the magnetic field of the

inductor Energy is transferred from the electric field

to the magnetic field

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Oscillations in an LCCircuit, 4 The capacitor becomes fully discharged

It stores no energy

All of the energy is stored in the magneticfield of the inductor

The current reaches its maximum value

The current now decreases inmagnitude, recharging the capacitorwith its plates having opposite theirinitial polarity

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Oscillations in an LCCircuit,

final Eventually the capacitor becomes fully

charged and the cycle repeats

The energy continues to oscillatebetween the inductor and the capacitor

The total energy stored in the LC circuit

remains constant in time and equals2

21

2 2IC L

QU U U L

C

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LCCircuit Analogy to Spring-

Mass System, 1

The potential energy kx2 stored in the spring isanalogous to the electric potential energy C(V

max

)2

stored in the capacitor All the energy is stored in the capacitor at t= 0 This is analogous to the spring stretched to its

amplitude

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Mass System, 2

The kinetic energy ( mv2) of the spring is analogousto the magnetic energy ( LI2) stored in theinductor

At t= T, all the energy is stored as magneticenergy in the inductor

The maximum current occurs in the circuit This is analogous to the mass at equilibrium

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Mass System, 3

At t= T, the energy in the circuit is

completely stored in the capacitor The polarity of the capacitor is reversed

This is analogous to the spring stretched to -A

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Mass System, 4

At t = T, the energy is again stored in the

magnetic field of the inductor This is analogous to the mass again reaching

the equilibrium position

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Active Figure 32.17


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Time Functions of an LC

Circuit In an LC circuit, charge can be expressed as

a function of time

Q = Qmax cos (t+ ) This is for an ideal LCcircuit

The angular frequency, , of the circuit

depends on the inductance and the

capacitance It is the natural frequencyof oscillation of the

circuit

1LC

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Time Functions of an LC

Circuit, 2 The current can be expressed as a

function of time

The total energy can be expressed as a

function of time

max I sin( )dQ

Q t dt

2

2 2 21

2 2

maxmaxcos I sinC L

QU U Ut L t

c

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Charge and Current in an LC

Circuit The charge on the

capacitor oscillates

between Qmax

and

-Qmax The current in the

inductor oscillates

betweenImax and -Imax Q andIare 90o out of

phase with each other So when Q is a maximum, I

is zero, etc.

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Energy in an LCCircuit

Graphs The energy

continually oscillatesbetween the energystored in the electricand magnetic fields

When the totalenergy is stored inone field, the energystored in the otherfield is zero

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Notes About Real LCCircuits In actual circuits, there is always some

resistance

Therefore, there is some energy transformedto internal energy

Radiation is also inevitable in this type of

circuit

The total energy in the circuit continuously

decreases as a result of these processes

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The RLCCircuit A circuit containing a

resistor, an inductor

and a capacitor iscalled an RLC

Circuit

Assume the resistor

represents the totalresistance of the

circuit

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Active Figure 32.21


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RLCCircuit, Analysis The total energy is not constant, since

there is a transformation to internal

energy in the resistor at the rate ofdU/dt= -I2R Radiation losses are still ignored

The circuits operation can beexpressed as

2

20

d Q dQ QL R

dt dt C

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RLCCircuit Compared to

Damped Oscillators The RLCcircuit is analogous to a

damped harmonic oscillator

When R= 0 The circuit reduces to an LCcircuit and is

equivalent to no damping in a mechanical

oscillator

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Damped Oscillators, cont. When Ris small:

The RLCcircuit is analogous to light

damping in a mechanical oscillator Q = Qmaxe

-Rt/2L cos dt

d is the angular frequency of oscillation

for the circuit and1

2 2

1

2d

R

LC L

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Damped Oscillators, final When Ris very large, the oscillations damp

out very rapidly There is a critical value ofRabove which no

oscillations occur

IfR= RC, the circuit is said to be critically

damped When R> RC, the circuit is said to be

overdamped

4 /CR L C

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Damped RLCCircuit, Graph The maximum value

ofQ decreases after

each oscillation R< RC

This is analogous to

the amplitude of a

damped spring-mass system

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Overdamped RLCCircuit,

Graph The oscillations

damp out very

rapidly Values ofRare

greater than RC

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