Self-Inductance RL Circuits Energy in a Magnetic Field Mutual Inductance Oscillations in an LC Circuit

When 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.

dt

dN B

L

E

BB

IB

dt

dIL E

dt

dIL

dt

dN B

L

E

I

NL B

dtdIL LE

(Henry=H=V.s/A)

Il

NnIB 00

Il

NABAB 0

l

AN

I

NL B

2

0

VnL 2

0

AlV

nlN

l N

A

Inductors 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.

dt

dILL E

0dt

dILIRE

/1 teR

I E

R

L

When S2 is at a

0dt

dILIR

/t

ieII R

L

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.

dt

dILIRII 2E

0dt

dILIRE

Battery

Power

Resistor

Power

Inductor

Power

The energy stored in the inductor

dt

dILI

dt

dU

II

IdILLIdIdUU00

2

2

1LIU

For energy density,

consider a solenoid

AlnL 2

0

nIB 0

AlB

n

BAlnLIU

0

22

0

2

0

2

22

1

2

1

0

2

2

B

Al

UuB

BdAB

a

bIl

r

drIlldr

r

IBdA

b

a

b

a

B ln222

000

ldrdA

a

bl

IL B ln

2

0

2

2

1LIU

a

blIU ln

4

2

0

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

1

12212

I

NM

dt

dIM

N

IM

dt

dN

dt

dN 1

12

2

1122

1222

E

dt

dIM 2

211 E

It can be shown that: MMM 2112

dt

dIM 2

1 Edt

dIM 1

2 E

Il

NB B

0

l

ANN

I

BAN

I

NM BHHBHH

0

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.

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

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.

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.

At some arbitrary time, t:

22

2

1

2LI

C

QUUU LC

0dt

dU

02

2

dt

QdL

C

Q

LC

1 Natural frequency of oscillation

tQQ cosmax

tItQI sinsin maxmax

tLI

tC

QUUU LC 2max

22

2

max sin2

cos2

22

max22

max LI

C

Q

ttC

QU 22

2

max sincos2

C

QU

2

2

max

a) f = ? Hz

LCf 6

12310

1091081.22

1

2

1

2

b) Qmax = ? and Imax = ?

CCQ 1012

max 1008.112109 E

AfQQI 4

maxmaxmax 1079.62

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

tCtQQ 610

max 102cos1008.1cos

tAtII 64

max 102sin1079.6sin

d) U = ? JC

QU 10

2

max 1048.62

RLC Circuit - A more realistic circuit

The resistor represents the losses in the system.

Can be oscillatory, but the amplitude decreases.

212

2

max

2

1

cos

L

R

LC

teQQ

d

d

LRt

CLR 4

Reading Assignment

Chapter 33 - Alternating Current Circuits

WebAssign: Assignment 10