Chapter 32 Inductance. Joseph Henry 1797 – 1878 1797 – 1878 American physicist American...

Chapter 32

InductanceInductance

Joseph Henry

1797 – 18781797 – 1878 American physicistAmerican physicist First director of the First director of the

SmithsonianSmithsonian Improved design of Improved design of

electromagnetelectromagnet Constructed one of the first Constructed one of the first

motorsmotors Discovered self-inductanceDiscovered self-inductance Unit of inductance is named Unit of inductance is named

in his honorin his honor

Some TerminologySome Terminology

Use Use emfemf and and currentcurrent when they are caused when they are caused by batteries or other sourcesby batteries or other sources

Use Use induced emfinduced emf and and induced currentinduced current when when they are caused by changing magnetic fieldsthey are caused by changing magnetic fields

When dealing with problems in When dealing with problems in electromagnetism, it is important to electromagnetism, it is important to distinguish between the two situationsdistinguish between the two situations

Self-InductanceSelf-Inductance

When the switch is When the switch is closed, the current closed, the current does not immediately does not immediately reach its maximum reach its maximum value value

Faraday’s law can be Faraday’s law can be used to describe the used to describe the effecteffect

R/

Self-InductanceSelf-Inductance As the current increases with time, the magnetic As the current increases with time, the magnetic

flux through the circuit loop due to this current flux through the circuit loop due to this current also increases with timealso increases with time

This increasing flux creates an This increasing flux creates an induced emf induced emf in in the circuitthe circuit


The direction of the The direction of the induced emf induced emf is such that it is such that it would cause an would cause an induced current induced current in the loop in the loop which would establish a magnetic field opposing which would establish a magnetic field opposing the change in the original magnetic fieldthe change in the original magnetic field

The direction of the The direction of the induced emf induced emf is opposite the is opposite the direction of the direction of the emfemf of the battery of the battery

This results in a gradual rather the This results in a gradual rather the instantaneous increase in the current to its final instantaneous increase in the current to its final equilibrium valueequilibrium value


This effect is called This effect is called self-inductanceself-inductance Because the changing flux through the circuit and Because the changing flux through the circuit and

the resultant the resultant induced emf induced emf arise from the circuit arise from the circuit itselfitself

The emf The emf εεLL is called a is called a self-induced emfself-induced emf


An An induced emf induced emf is always proportional to the time rate is always proportional to the time rate of change of the currentof change of the current The The emf emf is proportional to the flux, which is proportional to is proportional to the flux, which is proportional to

the field and the field is proportional to the currentthe field and the field is proportional to the current

LL is a constant of proportionality called the is a constant of proportionality called the inductanceinductance of the coil and it depends on the of the coil and it depends on the geometry of the coil and other physical characteristicsgeometry of the coil and other physical characteristics

L

d Iε L

dt

Inductance of a CoilInductance of a Coil

A closely spaced coil of A closely spaced coil of NN turns carrying turns carrying current current II has an inductance of has an inductance of

The inductance is a measure of the The inductance is a measure of the opposition to a change in currentopposition to a change in current

B LN εL

I d I dt

Inductance UnitsInductance Units

The SI unit of inductance is the The SI unit of inductance is the henryhenry ( (HH))

Named for Joseph HenryNamed for Joseph Henry

AsV

1H1

Inductance of a SolenoidInductance of a Solenoid

Assume a uniformly wound solenoid having Assume a uniformly wound solenoid having NN turns and length turns and length ℓℓ Assume Assume ℓℓ is much greater than the radius of the is much greater than the radius of the

solenoidsolenoid The flux through each turn of area The flux through each turn of area A isis

B o o

NBA μ nI A μ I A

InductanceInductance of a Solenoid

The inductance isThe inductance is

This shows that This shows that LL depends on the geometry depends on the geometry of the objectof the object

2oB μ N AN

LI

An inductor in the form of a solenoid contains An inductor in the form of a solenoid contains 420 420 turnsturns, is , is 16.0 cm 16.0 cm in length, and has a cross-sectional in length, and has a cross-sectional area of area of 3.00 cm3.00 cm22. What uniform rate of decrease of . What uniform rate of decrease of current through the inductor induces an emf of current through the inductor induces an emf of 175 175 μμVV? ?

An inductor in the form of a solenoid contains An inductor in the form of a solenoid contains 420 420 turnsturns, is , is 16.0 cm 16.0 cm in length, and has a cross-sectional in length, and has a cross-sectional area of area of 3.00 cm3.00 cm22. What uniform rate of decrease of . What uniform rate of decrease of current through the inductor induces an emf of current through the inductor induces an emf of 175 175 μμVV? ?

2 420 40

420 3.00 104.16 10 H

0.160N A

L

6

4

175 10 V0.421 A s

4.16 10 H

dI dILdt dt L

The current in a The current in a 90.0-mH90.0-mH inductor changes with inductor changes with time as time as I I = 1.00= 1.00tt22 – 6.00 – 6.00t t (in SI units). Find the (in SI units). Find the magnitude of the induced emf at (a) magnitude of the induced emf at (a) t t = 1.00 s = 1.00 s and and (b) (b) t t = 4.00 s. = 4.00 s. (c) At what time is the emf zero? (c) At what time is the emf zero?

The current in a The current in a 90.0-mH90.0-mH inductor changes with inductor changes with time as time as I I = 1.00= 1.00tt22 – 6.00 – 6.00t t (in SI units). Find the (in SI units). Find the magnitude of the induced emf at (a) magnitude of the induced emf at (a) t t = 1.00 s = 1.00 s and and (b) (b) t t = 4.00 s. = 4.00 s. (c) At what time is the emf zero? (c) At what time is the emf zero?

3 290.0 10 6 VdI d

L t tdt dt

(a) At , t=1s

(b) At , t= 4s

(c) when t = 3 sec .

360 mV

180 mV

390.0 10 2 6 0t

RL Circuit, IntroductionRL Circuit, Introduction

A circuit element that has a large self-A circuit element that has a large self-inductance is called an inductance is called an inductorinductor

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

the circuit is negligible compared to the the circuit is negligible compared to the inductorinductor However, even without a coil, a circuit will have However, even without a coil, a circuit will have

some self-inductancesome self-inductance

Effect of an Inductor in a CircuitEffect of an Inductor in a Circuit

The inductance results in a The inductance results in a back emfback emf Therefore, the inductor in a circuit opposes Therefore, the inductor in a circuit opposes

changes in current in that circuitchanges in current in that circuit The inductor attempts to keep the current the The inductor attempts to keep the current the

same way it was before the change occurredsame way it was before the change occurred The inductor can cause the circuit to be “sluggish” The inductor can cause the circuit to be “sluggish”

as it reacts to changes in the voltageas it reacts to changes in the voltage

RLRL Circuit, Analysis Circuit, Analysis An An RLRL circuit contains an circuit contains an

inductor and a resistorinductor and a resistor Assume Assume SS22 is connected to is connected to aa

When switchWhen switch S S11 is closed (at is closed (at

time time tt = 0 = 0), the current begins ), the current begins to increaseto increase

At the same time, a back emf At the same time, a back emf is induced in the inductor that is induced in the inductor that opposes the original opposes the original increasing currentincreasing current

RLRL Circuit, Analysis Circuit, Analysis

Applying Kirchhoff’s loop rule to the circuit in Applying Kirchhoff’s loop rule to the circuit in the clockwise direction givesthe clockwise direction gives

Looking at the current, Looking at the current, we findwe find

0d I

ε I R Ldt

1 Rt LεI e

R

RLRL Circuit, Analysis Circuit, Analysis

The inductor affects the current exponentiallyThe inductor affects the current exponentially The current does not instantly increase to its The current does not instantly increase to its

final equilibrium valuefinal equilibrium value If there is no inductor, the exponential term If there is no inductor, the exponential term

goes to zero and the current would goes to zero and the current would instantaneously reach its maximum value as instantaneously reach its maximum value as expectedexpected

RL Circuit, Time ConstantRL Circuit, Time Constant

The expression for the current can also be The expression for the current can also be expressed in terms of the time constant, expressed in terms of the time constant, , of , of the circuitthe circuit

where where = L / R = L / R Physically, Physically, is the time required for the is the time required for the

current to reach current to reach 63.2%63.2% of its maximum value of its maximum value

1 t τεI e

R

In the circuit shown in Figure In the circuit shown in Figure L L = 7.00 H= 7.00 H, ,

R R = 9.00 Ω= 9.00 Ω, and , and εε = 120 V = 120 V. What is the self-induced . What is the self-induced emf emf 0.200 s0.200 s after the switch is closed? after the switch is closed?

In the circuit shown in Figure In the circuit shown in Figure L L = 7.00 H= 7.00 H, ,

R R = 9.00 Ω= 9.00 Ω, and , and εε = 120 V = 120 V. What is the self-induced . What is the self-induced emf emf 0.200 s0.200 s after the switch is closed? after the switch is closed?

1.80 7.001201 1 3.02 A

9.00tI e e

R

3.02 9.00 27.2 V

120 27.2 92.8 V

R

L R

V IR

V V

RLRL Circuit, Current-Time Graph Circuit, Current-Time Graph The equilibrium value The equilibrium value

of the current is of the current is //RR and is reached as and is reached as tt approaches infinityapproaches infinity

The current initially The current initially increases very rapidlyincreases very rapidly

The current then The current then gradually approaches gradually approaches the equilibrium valuethe equilibrium value

RLRL Circuit, Current-Time Graph Circuit, Current-Time Graph

The time rate of change The time rate of change of the current is a of the current is a maximum at maximum at tt = 0 = 0

It falls off exponentially It falls off exponentially as as tt approaches infinityapproaches infinity

In general, In general,

t τd I εe

dt L

RL Circuit Without A BatteryRL Circuit Without A Battery

Now set Now set SS22 to position to position bb The circuit now contains The circuit now contains

just the right hand loop just the right hand loop The battery has been The battery has been

eliminatedeliminated The expression for the The expression for the

current becomescurrent becomes

t tτ τ

i

εI e I e

R

A A 10.0-mH10.0-mH inductor carries a current inductor carries a current I I = = IImaxmax sin sin ωtωt, ,

with with IImaxmax = 5.00 A = 5.00 A and and ωω/2π = 60.0 Hz/2π = 60.0 Hz. What is the . What is the

back emf as a function of time? back emf as a function of time?

A A 10.0-mH10.0-mH inductor carries a current inductor carries a current I I = = IImaxmax sin sin ωtωt, ,

with with IImaxmax = 5.00 A = 5.00 A and and ωω/2π = 60.0 Hz/2π = 60.0 Hz. What is the . What is the

back emf as a function of time? back emf as a function of time?

3back max maxsin cos 10.0 10 120 5.00 cos

dI dL L I t L I t tdt dt

back 6.00 cos 120 18.8 V cos 377t t

Energy in a Magnetic FieldEnergy in a Magnetic Field

In a circuit with an inductor, the battery must In a circuit with an inductor, the battery must supply more energy than in a circuit without supply more energy than in a circuit without an inductoran inductor

Part of the energy supplied by the battery Part of the energy supplied by the battery appears as internal energy in the resistorappears as internal energy in the resistor

The remaining energy is stored in the The remaining energy is stored in the magnetic field of the inductormagnetic field of the inductor


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

II is the rate at which energy is being supplied by is the rate at which energy is being supplied by the batterythe battery

II22RR is the rate at which the energy is being delivered is the rate at which the energy is being delivered to the resistorto the resistor

Therefore, Therefore, LI (dI/dt) LI (dI/dt) must be the rate at which the must be the rate at which the energy is being stored in the inductorenergy is being stored in the inductor

2 d II ε I R LI

dt


Let Let UU denote the energy stored in the denote the energy stored in the inductor at any timeinductor at any time

The rate at which the energy is stored isThe rate at which the energy is stored is

To find the total energy, integrate andTo find the total energy, integrate and

dU d ILI

dt dt

2

0

1

2

IU L I d I LI

Energy Density of a Magnetic Energy Density of a Magnetic FieldField

Given Given U = ½ L IU = ½ L I22 and assume (for simplicity) a solenoid and assume (for simplicity) a solenoid with with L = L = oo n n22 V V

Since Since V V is the volume of the solenoid, the magnetic is the volume of the solenoid, the magnetic energy density, energy density, uuBB is is

This applies to any region in which a magnetic field exists This applies to any region in which a magnetic field exists (not just the solenoid)(not just the solenoid)

2 221

2 2oo o

B BU μ n V V

μ n μ

2

2Bo

U Bu

V μ

Energy Storage SummaryEnergy Storage Summary

A resistor, inductor and capacitor all store A resistor, inductor and capacitor all store energy through different mechanismsenergy through different mechanisms Charged capacitor Charged capacitor

Stores energy as electric potential energyStores energy as electric potential energy InductorInductor

When it carries a current, stores energy as magnetic When it carries a current, stores energy as magnetic potential energypotential energy

ResistorResistor Energy delivered is transformed into internal energyEnergy delivered is transformed into internal energy

The magnetic field inside a superconducting The magnetic field inside a superconducting solenoid is solenoid is 4.50 T4.50 T. The solenoid has an inner . The solenoid has an inner diameter of diameter of 6.20 cm 6.20 cm and a length of and a length of 26.0 cm26.0 cm. . Determine (a) the magnetic energy density in the Determine (a) the magnetic energy density in the field and (b) the energy stored in the magnetic field field and (b) the energy stored in the magnetic field within the solenoid. within the solenoid.

The magnetic field inside a superconducting The magnetic field inside a superconducting solenoid is solenoid is 4.50 T4.50 T. The solenoid has an inner . The solenoid has an inner diameter of diameter of 6.20 cm 6.20 cm and a length of and a length of 26.0 cm26.0 cm. . Determine (a) the magnetic energy density in the Determine (a) the magnetic energy density in the field and (b) the energy stored in the magnetic field field and (b) the energy stored in the magnetic field within the solenoid. within the solenoid.

(a) The magnetic energy density is given by

226 3

60

4.50 T8.06 10 J m

2 2 1.26 10 T m A

B

.

(b) The magnetic energy stored in the field equals u times the volume of the solenoid (the volume in which B is non-zero).

26 38.06 10 J m 0.260 m 0.0310 m 6.32 kJU uV

Example: The Coaxial CableExample: The Coaxial Cable

Calculate Calculate LL for the for the cablecable

The total flux isThe total flux is

Therefore, Therefore, LL is is

2

ln2

bo

B a

o

μ IB dA dr

πrμ I b

π a

ln2

oB μ bL

I π a

Mutual InductanceMutual Inductance

The magnetic flux through the area enclosed The magnetic flux through the area enclosed by a circuit often varies with time because of by a circuit often varies with time because of time-varying currents in nearby circuitstime-varying currents in nearby circuits

This process is known as This process is known as mutual inductionmutual induction because it depends on the interaction of two because it depends on the interaction of two circuitscircuits


The current in The current in coil 1 coil 1 sets up a magnetic fieldsets up a magnetic field

Some of the magnetic Some of the magnetic field lines pass through field lines pass through coil 2coil 2

Coil 1 has a current Coil 1 has a current II1 1

and and NN11 turnsturns Coil 2 has Coil 2 has NN22 turnsturns


The The mutual inductancemutual inductance MM1212 of of coil 2coil 2 with with

respect to respect to coil 1coil 1 is is

Mutual inductance depends on the geometry Mutual inductance depends on the geometry of both circuits and on their orientation with of both circuits and on their orientation with respect to each otherrespect to each other

2 1212

1

NM

I

Induced emf in Mutual Induced emf in Mutual InductanceInductance

If current If current II11 varies with time, the emf induced varies with time, the emf induced

by by coil 1coil 1 in in coil 2coil 2 is is

If the current is in If the current is in coil 2coil 2, there is a mutual , there is a mutual inductance inductance MM2121

If If current 2 current 2 varies with time, the emf induced varies with time, the emf induced by by coil 2coil 2 in in coil 1coil 1 is is

12 12 2 12

d d Iε N M

dt dt

21 21

d Iε M

dt


In mutual induction, the emf induced in one In mutual induction, the emf induced in one coil is always proportional to the rate at which coil is always proportional to the rate at which the current in the other coil is changingthe current in the other coil is changing

The mutual inductance in one coil is equal to The mutual inductance in one coil is equal to the mutual inductance in the other coilthe mutual inductance in the other coil MM1212 = M = M2121 = M = M

The induced emf’s can be expressed asThe induced emf’s can be expressed as

2 11 2and

d I d Iε M ε M

dt dt

Two coils are close to each other. The first Two coils are close to each other. The first coil carries a time-varying current given by coil carries a time-varying current given by II((tt) = (5.00 A) ) = (5.00 A) ee–0.0250 –0.0250 tt sin(377sin(377tt)).. At At t t = 0.800 s= 0.800 s, the emf measured across the , the emf measured across the second coil is second coil is –3.20 V–3.20 V. What is the mutual . What is the mutual inductance of the coils? inductance of the coils?

Two coils are close to each other. The first coil Two coils are close to each other. The first coil carries a time-varying current given by carries a time-varying current given by II((tt) = (5.00 A) ) = (5.00 A) ee–0.0250 –0.0250 tt sin(377sin(377tt)).. At At t t = 0.800 s= 0.800 s, the , the emf measured across the second coil is emf measured across the second coil is –3.20 V–3.20 V. . What is the mutual inductance of the coils? What is the mutual inductance of the coils?

1 max sintI t I e t max 5.00 AI 10.0250 s

377 rad s 1max sin costdII e t t

dt

0.800 st

0.02001 5.00 A s 0.0250 sin 0.800 377 377cos 0.800 377dI

edt

31 1.85 10 A sdIdt

At

,

.

12

dIM

dt

23

1

3.20 V1.73 mH

1.85 10 A sM

dI dt

:

.

LCLC Circuits Circuits

A capacitor is A capacitor is connected to an connected to an inductor in an inductor in an LC LC circuitcircuit

Assume the capacitor Assume the capacitor is initially charged and is initially charged and then the switch is then the switch is closedclosed

Assume no resistance Assume no resistance and no energy losses and no energy losses to radiationto radiation

Oscillations in an Oscillations in an LCLC Circuit Circuit

Under the previous conditions, the current in Under the previous conditions, the current in the circuit and the charge on the capacitor the circuit and the charge on the capacitor oscillate between maximum positive and oscillate between maximum positive and negative valuesnegative values

With zero resistance, no energy is With zero resistance, no energy is transformed into internal energytransformed into internal energy

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

radiationradiation


The capacitor is fully chargedThe capacitor is fully charged The energy The energy UU in the circuit is stored in the electric in the circuit is stored in the electric

field of the capacitorfield of the capacitor The energy is equal to The energy is equal to QQ22

maxmax / 2C / 2C The current in the circuit is zeroThe current in the circuit is zero No energy is stored in the inductorNo energy is stored in the inductor

The switch is closedThe switch is closed


The current is equal to the rate at which the The current is equal to the rate at which the charge changes on the capacitorcharge changes on the capacitor As the capacitor discharges, the energy stored in As the capacitor discharges, the energy stored in

the electric field decreasesthe electric field decreases Since there is now a current, some energy is Since there is now a current, some energy is

stored in the magnetic field of the inductorstored in the magnetic field of the inductor Energy is transferred from the electric field to the Energy is transferred from the electric field to the

magnetic fieldmagnetic field


Eventually, the capacitor becomes fully Eventually, the capacitor becomes fully dischargeddischarged It stores no energyIt stores no energy All of the energy is stored in the magnetic field of All of the energy is stored in the magnetic field of

the inductorthe inductor The current reaches its maximum valueThe current reaches its maximum value

The current now decreases in magnitude, The current now decreases in magnitude, recharging the capacitor with its plates recharging the capacitor with its plates having opposite their initial polarityhaving opposite their initial polarity


The capacitor becomes fully charged and the The capacitor becomes fully charged and the cycle repeatscycle repeats

The energy continues to oscillate between The energy continues to oscillate between the inductor and the capacitorthe inductor and the capacitor

The total energy stored in the The total energy stored in the LC LC circuit circuit remains constant in time and equalsremains constant in time and equals

221

2 2IC L

QU U U L

C

LCLC Circuit Analogy to Spring- Circuit Analogy to Spring-Mass SystemMass System

The potential energy The potential energy ½kx½kx22 stored in the spring is analogous to stored in the spring is analogous to the electric potential energy the electric potential energy (Q(Qmaxmax))22/(2C) /(2C) stored in the stored in the

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


The kinetic energy The kinetic energy (½ mv(½ mv22) ) of the spring is analogous to the of the spring is analogous to the magnetic energy magnetic energy (½ L I(½ L I22) ) stored in the inductorstored in the inductor

At At t = ¼ Tt = ¼ T, all the energy is stored as magnetic energy in the , all the energy is stored as magnetic energy in the inductorinductor

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


At At t = ½ Tt = ½ T, the energy in the circuit is completely , the energy in the circuit is completely stored in the capacitorstored in the capacitor

The polarity of the capacitor is reversedThe polarity of the capacitor is reversed This is analogous to the spring stretched to This is analogous to the spring stretched to -A-A


At At t = ¾ Tt = ¾ T, the energy is again stored in the , the energy is again stored in the magnetic field of the inductormagnetic field of the inductor

This is analogous to the mass again reaching the This is analogous to the mass again reaching the equilibrium positionequilibrium position


At At tt = = TT, the cycle is completed, the cycle is completed The conditions return to those identical to the initial conditionsThe conditions return to those identical to the initial conditions At other points in the cycle, energy is shared between the At other points in the cycle, energy is shared between the

electric and magnetic fieldselectric and magnetic fields

Time Functions of an Time Functions of an LCLC CircuitCircuit In an In an LC LC circuit, charge can be expressed as circuit, charge can be expressed as

a function of timea function of time Q = QQ = Qmaxmax cos ( cos (ωωt + t + φφ)) This is for an ideal This is for an ideal LCLC circuit circuit

The angular frequency, The angular frequency, ωω, of the circuit , of the circuit depends on the inductance and the depends on the inductance and the capacitancecapacitance It is the natural frequency of oscillation of the It is the natural frequency of oscillation of the

circuitcircuit 1ωLC

Time Functions of an Time Functions of an LCLC Circuit Circuit

The current can be expressed as a function The current can be expressed as a function of timeof time

The total energy can be expressed as a The total energy can be expressed as a function of timefunction of time

max

dQI ωQ sin(ωt φ)

dt

22 2 21

2 2max

C L max

QU U U cos ωt LI sin ωt

c

Charge and Current in an Charge and Current in an LCLC CircuitCircuit

The charge on the capacitor The charge on the capacitor oscillates between oscillates between QQmaxmax and and

--QQmaxmax

The current in the inductor The current in the inductor oscillates between oscillates between IImaxmax and and

--IImaxmax

QQ and and II are are 9090oo out of phase out of phase with each otherwith each other So when So when QQ is a is a

maximum, maximum, II is zero, etc. is zero, etc.

Energy in an Energy in an LCLC Circuit – Graphs Circuit – Graphs

The energy continually The energy continually oscillates between the oscillates between the energy stored in the energy stored in the electric and magnetic electric and magnetic fieldsfields

When the total energy When the total energy is stored in one field, is stored in one field, the energy stored in the the energy stored in the other field is zeroother field is zero

Notes About Real Notes About Real LCLC Circuits Circuits

In actual circuits, there is always some In actual circuits, there is always some resistanceresistance

Therefore, there is some energy transformed Therefore, there is some energy transformed to internal energyto internal energy

Radiation is also inevitable in this type of Radiation is also inevitable in this type of circuitcircuit

The total energy in the circuit continuously The total energy in the circuit continuously decreases as a result of these processesdecreases as a result of these processes

The The RLCRLC Circuit Circuit

A circuit containing a A circuit containing a resistor, an inductor resistor, an inductor and a capacitor is and a capacitor is called an called an RLCRLC CircuitCircuit

Assume the resistor Assume the resistor represents the total represents the total resistance of the circuitresistance of the circuit

PLAYACTIVE FIGURE

Active Figure 32.15

Use the active figure to adjust R, L, and C. Observe the effect on the charge

PLAYACTIVE FIGURE

RLCRLC Circuit, Analysis Circuit, Analysis

The total energy is not constant, since there The total energy is not constant, since there is a transformation to internal energy in the is a transformation to internal energy in the resistor at the rate of resistor at the rate of dUdU//dtdt = - = -II22RR Radiation losses are still ignoredRadiation losses are still ignored

The circuit’s operation can be expressed asThe circuit’s operation can be expressed as

2

20

d Q dQ QL R

dt dt C

RLCRLC Circuit Compared to Circuit Compared to Damped OscillatorsDamped Oscillators

The The RLCRLC circuit is analogous to a damped circuit is analogous to a damped harmonic oscillatorharmonic oscillator

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

equivalent to no damping in a mechanical equivalent to no damping in a mechanical oscillatoroscillator


When When RR is small: is small: The The RLC RLC circuit is analogous to light damping in a circuit is analogous to light damping in a

mechanical oscillatormechanical oscillator Q = QQ = Qmaxmax e e-Rt/2L-Rt/2L cos cos ωωddtt

ωωdd is the angular frequency of oscillation for the is the angular frequency of oscillation for the

circuit and circuit and 1

2 21

2d

Rω

LC L


When When R R is very large, the oscillations damp out very is very large, the oscillations damp out very rapidlyrapidly

There is a critical value of There is a critical value of RR above which no above which no oscillations occuroscillations occur

If If R = RR = RCC, the circuit is said to be , the circuit is said to be critically dampedcritically damped

When When R > RR > RCC, the circuit is said to be , the circuit is said to be overdampedoverdamped

4 /CR L C

Damped Damped RLCRLC Circuit, Graph Circuit, Graph

The maximum value of The maximum value of QQ decreases after each decreases after each oscillationoscillation RR < < RRCC

This is analogous to the This is analogous to the amplitude of a damped amplitude of a damped spring-mass systemspring-mass system

Summary: Analogies Between Electrical and Mechanic Systems

Chapter 32 Inductance. Joseph Henry 1797 – 1878 1797 – 1878 American physicist American...

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Transcript of Chapter 32 Inductance. Joseph Henry 1797 – 1878 1797 – 1878 American physicist American...