Lectures 18-19 (Ch. 30)Inductance and Self-inductunce

1. Mutual inductunce2. Tesla coil3. Inductors and self-inductance4. Toroid and long solenoid5. Inductors in series and parallel6. Energy stored in the inductor, energy density7. LR circuit8. LC circuit9. LCR circuit

Mutual inductanceVirce verse: if current in coil 2 is changing, the changing flux through coil 1 induces emf in coil 1.

dt

diM

i

NM

iMNdt

dN

1212

1

2221

12122

222

1

22

2

11

1212

i

N

i

NM

MMM

dt

diM

i

NM

iMNdt

dN

2121

2

1112

21211

111

Units of M

sH

m

V

C

NE

m

V

q

FE

m

Vs

Cm

NsTBvqF

sA

Vs

A

mTH

henryHA

WbM

iM

1

][,

111

1

111

)(11

1][,

2

2

Joseph Henry (1797-1878)

Typical magnitudes: 1μH-1mH

Examples where mutual inductance is useful

Tesla coil

Nikola Tesla (1856 –1943)[B]=1T to his honor

?

,,.1

2

2110

1

2110

1

1101101

1121

2212

l

NNAMwhy

l

NNAM

l

iNinB

ABi

NMrr

Estimate.

kVmHMKtake

Kl

NNAM

coreticferromagneWith

Vs

AH

dt

diM

HmA

mTmM

ts

AiNNmcmAml

m

m

50,251000

,

501021025

255.0

101010104

102,10,1000,1010,5.0

1

01

211

6621

3237

6221

23211

cos

.3

.2

1

2120

2

1

2120

21212

l

NNAM

axisesbetweenangleanisandrrif

l

NNAM

ABrrif

Example: M=?

Mutual inductance may induce unwanted emf in nearby circuits. Coaxial cables are used to avoid it.

Self-inductance

i

NL

dt

diL

LiNdt

dN

,

,

Thin ToroidThin solenoid with approximately equal inner and outer radius.

r

ANL

r

NiB

BAi

NL

2

2

,

2

Long solenoid

l

ANL

l

NiB

ABi

NL

2

max

max

maxmax,

Example. Toroidal solenoid with a rectangular area.

a

bhNL

a

bNih

r

drNih

r

NiB

Bhdrdi

NL

b

a

ln2

ln22

,2

,

2

dt

diLV

dt

diL

ab

Inductors in circuits

Energy stored in inductor

2

22

2

0

LIU

LILidiW

LididtPdWdt

diiLIVP

L

I

in

abin

222

22 QVCV

C

QUC Compare to

Magnetic energy density

Let’s consider a thin toroidal

solenoid, but the result turns out to be correct for a general case

rAr

AINu

r

ANL

LIU

rA

U

volume

Uu

B

L

LLB

222

2,

2

2

22

22

2

222

)2(

2

2

2

22

222

Bu

NrAr

rABNu

N

rBI

r

NIB

B

B

Energy is stored in B inside the inductor

Compare to:

Energy is stored in E inside the capacitor

2

2EuE

Example. Find U of a toroidal solenoid with rectangular area

4

ln

2

2,

2

.2

4

ln

ln2

,2

.1

22

2

22

22

ab

hNIU

rdrhdV

r

NIB

Bu

dVuU

ab

hNIU

a

bhNL

LIU

L

B

volune

BL

L

L

LR circuit, storing energy in the inductor

t

L

t

edt

diL

RIeIi

t

R

Ri

R

Ldt

Ri

diL

R

Ri

dt

didt

diLiR

),1(

))(

ln(

,)(

)(

0

εL

Energy conservation law

dt

dURii

LiU

dt

dU

dt

dLi

dt

diLi

dt

diLiRii

dt

diLiR

L

LL

2

22

2

2,

2

1

,,0

Power output of the battery =power dissipated in the resistor + the rate at which the energy is stored in inductor

General solution Initial conditions (t=0)

L

i 0

t

L

t

edt

diL

RIeIi

),1(

0

L

RIi

Steady state (t→∞)

LR circuit, delivering energy from inductor

ttt

L

t

i

I

i

I

eRIeeIL

dt

diL

Ieit

I

i

R

Ldt

i

di

L

Ri

dt

di

dt

diLiR

,,ln

,

,0

Ridt

dURi

dt

Lid

Ridt

idiL

dt

diLiR

L 222

2

,)2/(

0,0

εL

t

ε

The rate of energy decrease in inductor is equal to the power input to the resistor.

Oscillations in LC circuit

Oscillations in LC circuit

,0,,02

2

LC

q

dt

qd

dt

dqi

dt

diL

C

q

conditionsinitialthebydefinedareQand

tQi

tQq

)sin(

)cos(

LCqq

1,0 22

Compare to mechanical oscillator

x0

F

m

kxx

kxxm

,0

,

2

LCm

k

CkLm

Limv

C

qkx

ivqx

1,

1,

22,

22

,2222

conditionsinitialbydefinedareXand

tXv

tXx

)sin(

)cos(

General solution

LCqq

1,0 22

tqi

tqq

qQtqtQq

ti

tiqtq

sin

cos

)0(cos

00)0(

0)0(,)0(.1

0

0

0

0

q

i

t

t

conditionsinitialbydefinedareQand

2

T

2

T

ti

q

tii

iQiQti

tQtQi

tQtQq

Qtq

tQi

tQq

solutiongeneral

ititq

sin

cos

)0(

cos)2/sin(

sin)2/cos(

2/0cos)0(

)sin(

)cos(

)0(,0)0(.2

0

0

00

0

q

i

t

t2

T

2

T

2

202

02

20

20222

02

0

0

0

0

0

0

0

00

)()()(

tantan

sin

cos

)sin(

)cos(

)0(,)0(:.3

iq

iqQQiq

q

iar

q

i

Qi

Qq

tQi

tQq

itiqtqconditionsinitialArbitrary

Energy conservation law

0)2

()2(

0

,0

22

Li

dt

d

C

q

dt

d

dt

diLi

dt

dq

C

qdt

dqi

dt

diL

C

q

222222

2220

20

22 LI

C

QLi

C

qconst

Li

C

q

t

UL UC

q-Q Q

UC ULUC+UL=const

T/22

T

0)2/cos()4/cos()3

1012.322

.2

005.0)0(cos

00)0(

0)0(,)0(.1

00

620

2

maxmax

0

0

qTqq

JC

q

C

QUU

mCqQtqtQq

ti

tiqtq

CL

Example. In LC circuit C=0.4 mF, L=0.09H.The initial charge on the capacitor is 0.005mC and the initial current is zero. Find: (a) Maximum charge in the capacitor (b) Maximum energy stored in the inductor; (c) the charge at the moment t=T/4, where T is a period of oscillations.

mVC

q

dt

diLV

TtmVC

Lti

C

LCtiV

LCtiI

QC

QV

4)2

4/,4.2)0()0(

)0(,)1

Example. In LC circuit C=250 ϻF, L=60mH.The initial current is 1.55 mA and the initial charge is zero. 1) Find the maximum voltage across the capacitor . At which moment of time (closest to an initial moment) it is reached? 2) What is a voltage across an inductor when a charge on the capacitor is 1 ϻ C?

2

T

q

Cdt

diCLq

dt

diL

C

qd

LLLJ

iL

C

qc

mCqb

AiAT

Tiiia

ef

ef

108)

111,1025.2

22)

4.0)3()

1.0)3(,1.0)

2()

2()

3()

11

11

21

2

20

20

2111

Example. In LC circuit C=18 ϻF, two inductors are placed in parallel: L1=L2=1.5H and mutual inductance is negligible.The initial charge on the capacitor is 0.4mC and the initial current through the capacitor is 0.2A. Find: (a) the current in each inductor at the instant t=3π/ω, where ω is an eigen frequency of oscillations; (b) what is the charge at the same instant? (c) the maximum energy stored in the capacitor;(d) the charge on the capacitor when the current in each inductor is changing at a rate of 3.4 A/s.

LCR circuit

0

,0

C

qqRqL

dt

dqiiR

dt

diL

C

q

LCL

Rqqq

1,

2,02 0

20

teq ~ 02 20

2 Characteristic equation

C

LR

L

R

LC

4

4

10

,)(

22

222

02

220

22202,1

Critical damping

a) Underdamped oscillations:

tQetQ

tttQti

ttQtq

i

)(

)],sin()cos()[()(

)cos()()(2,1

2220

4R

C

L

2,0

)(

)(

210

2211

21

2,1

21

21

if

eCeCti

eCeCtq

numbersrealare

tt

tt

2220

4R

C

Lc) Overdamped oscillations:

t

t

eCtCCti

etCCtq

RC

L

2

1

])([)(

)()(

4

221

21

2,1

2220

b) Critically damped oscillations: 2220

4R

C

L

Example. The capacitor is initially uncharged. The switch starts in the open position and is then flipped to position 1 for 0.5s. It is then flipped to position 2 and left there.1) What is a current through the coil at the moment t=0.5s (i.e. just before the switch was flipped to position 2)?2) If the resistance is very small, how much electrical energy will be dissipated in it?3) Sketch a graph showing the reading of the ammeter as a function of time after the switch is in position 2, assuming that r is small.

1 2

r

25Ω

10mH50V

AIsti

smsH

R

L

AV

RIeIi

t

2)5.0(

5.04.025

10

225

50),1()1

2

mJLI

UU Ldis 202

)22

3)

10µF

A

Induced oscillations in LRC circuit, resonance

~

)(

2tan,

])2()[(

sin2

cos)(

]sin)sin(cos)[cos(

)cos()sin(2)cos()(

)cos(

)sin(

)cos(

cos2

,,2

,1

,

0cos

20

222220

22

220

220

2

20

0

fQ

fQ

fQ

ttf

tftQtQ

tQq

tQq

tQq

tfqqq

Lf

L

R

LCdt

dqi

tRidt

diL

C

q

Q

0At the resonance condition: an amplitude greatly insreases0

0