InductanceSelf-InductanceRL CircuitsEnergy in a Magnetic FieldMutual InductanceOscillations in an LC Circuit

Self-InductanceWhen the switch is closed, the battery (source emf) starts pushing electrons around the circuit.

The current starts to rise, creating an increasing magnetic flux through the circuit.

This increasing flux creates an induced emf in the circuit.

The induced emf will create a flux to oppose the increasing flux.

The direction of this induced emf will be opposite the source emf.

This results in a gradual increase in the current rather than an instantaneous one.

The induced emf is called the self-induced emf or back emf.

When I changes, an emf is induced in the coil.

If I is increasing (and therefore increasing the flux through the coil), then the induced emf will set up a magnetic field to oppose the increase in the magnetic flux in the direction shown.

If I is decreasing, then the induced emf will set up a magnetic field to oppose the decrease in the magnetic flux.

Current in a Coil

Self-Inductance

dtdN B

LΦ

−=E

BB ∝Φ

IB ∝

dtdI

L ∝E

dtdIL

dtdN B

L −=Φ

−=E

INL BΦ

=

dtdIL LE−=

(Henry=H=V.s/A)

Inductance of a Solenoid

IlNnIB 00 μμ ==

Il

NABAB 0μ==Φ

lAN

INL B

2

0μ=Φ

= VnL 20μ=

AlVnlN

==

lN

A

RL CircuitsInductors are circuit elements with large self-induction.A circuit with an inductor will generate some back-emf in response to a changing current.This back-emf will act to keep the current the way it used to be.Such a circuit will act “sluggish” in its response.

RL Circuit Analysis

dtdILL −=E

0=−−dtdILIRE

( )τ/1 teR

I −−=E

RL

=τ

When S2 is at a

0=+dtdILIR

τ/tieII −=

RL

=τ

When S2 is at b

When the switch is closed, the current through the circuit exponentially approaches a value I = E / R. If we repeat this experiment with an inductor having twice the number of turns per unit length, the time it takes for the current to reach a value of I / 2

1. increases.2. decreases.3. is the same.

Concept Question

Energy in a Magnetic Field

dtdILIRII += 2E

0=−−dtdILIRE

Battery Power

Resistor Power

Inductor Power

The energy stored in the inductor

dtdILI

dtdU

=

∫∫∫ ===II

IdILLIdIdUU00

2

21 LIU =

For energy density, consider a solenoid

AlnL 20μ=

nIB 0μ=

AlBn

BAlnLIU0

22

0

20

2

221

21

μμμ =⎟⎟

⎠

⎞⎜⎜⎝

⎛==

0

2

2μB

AlUuB ==

Inductance of a Coaxial Cable∫=Φ BdAB

⎟⎠⎞

⎜⎝⎛====Φ ∫∫∫ a

bIlrdrIlldr

rIBdA

b

a

b

aB ln

222000

πμ

πμ

πμ

ldrdA =

⎟⎠⎞

⎜⎝⎛=

Φ=

abl

IL B ln

20

πμ

2

21 LIU =

⎟⎠⎞

⎜⎝⎛=

ablIU ln

4

20

πμ

Mutual InductanceA change in the current of one circuit can induce an emf in a nearby circuit

1

12212 I

NM Φ=

dtdIM

NIM

dtdN

dtdN 1

122

1122

1222 −=⎟⎟

⎠

⎞⎜⎜⎝

⎛−=

Φ−=E

dtdIM 2

211 −=E

It can be shown that: MMM == 2112

dtdIM 2

1 −=EdtdIM 1

2 −=E

Wireless Battery Charger

Il

NB B0μ=

lANN

IBAN

INM BHHBHH

0μ==Φ

=

LC Oscillators

Provide a transfer of energy between the capacitor and the inductor.Analogous to a spring-block system

Assume the capacitor is fully charged and thus has some stored energy.

When the switch is closed, the charges on the capacitor leave the plates and move in the circuit, setting up a current.

The Oscillation Cycle

This current starts to discharge the capacitor and reduce its stored energy.

The Oscillation Cycle

When the capacitor is fully discharged, it stores no energy, but the current reaches a maximum and all the energy is stored in the inductor.

At the same time, the current increases the stored energy in the magnetic field of the inductor.

The Oscillation Cycle

When the capacitor becomes fully charged (with the opposite polarity), the energy is completely stored in the capacitor again.

The e-field sets up the current flow in the opposite direction.

The current continues to flow and now starts to charge the capacitor again, this time with the opposite polarity.

Then the cycle completes itself in reverse.

The Oscillation Cycle

Charge and Current in LC CircuitsAt some arbitrary time, t:

22

21

2LI

CQUUU LC +=+=

0=dtdU

02

2

=+dt

QdLCQ

LC1

=ω Natural frequency of oscillation

tQQ ωcosmax=

tItQI ωωω sinsin maxmax −=−=

tLItC

QUUU LC ωω 2max2

22

max sin2

cos2

+=+=

22max

22max LIC

Q=

( )ttC

QU ωω 222

max sincos2

+=

CQU2

2max=

Energy in LC Circuits

a) f=? ( )( ) HzLC

f 6

12310

1091081.221

21

2=

××==

−−πππω

b) Qmax =? and Imax =?

( )( ) CCQ 1012max 1008.112109 −− ×=×== E

AfQQI 4maxmaxmax 1079.62 −×=== πω

c) Q(t)=? and I(t)=?

( ) ( )tCtQQ 610max 102cos1008.1cos ××== − πω

( ) ( )tAtII 64max 102sin1079.6sin ××−=−= − πω

d) U=? JC

QU 102

max 1048.62

−×==

Oscillations in an LC Circuit

RLC Circuits –

Damped Oscillations

RLC Circuit - A more realistic circuitThe resistor represents the losses in the system.Can be oscillatory, but the amplitude decreases.

212

2max

21

cos

⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎠⎞

⎜⎝⎛−=

= −

LR

LC

teQQ

d

dLRt

ω

ω

CLR 4<

For Next ClassReading Assignment

Chapter 33 - Alternating Current Circuits

WebAssign: Assignment 10