Lesson 9 Lesson 9Lesson 9

Faraday’s LawFaraday’s Law

Faraday’s Law of InductionFaraday’s Law of InductionMotional EMFMotional EMFLenz’s LawLenz’s LawInduced EMF’s and Induced Electric FieldsInduced EMF’s and Induced Electric FieldsEddy CurrentsEddy Currents

Torque on LoopCurrent in loop in a magnetic

field produces torque on a loop

In general the torque on a current loop in magnetic field is

= M B

M IA, where A is the area vector of the loop,

with the orientation given by the current flow.

Induced CurrentDoes torque on loop in a magnetic field produces current in a loop ?

YESYES

Picture

current depends on the torquethus on rotational frequency

II

B

Change of Flux Picture

Current depends on speed of magnetThus rate of change of magnetic Field

Change of Flux Picture Equations

Common factors, change of area, change of magnetic field

Iind dBdt

Iind dA

dtA area perpendicular to magnetic Field

Iind dB

dtB A B

Induced Current in Wire

I

B v

FB

moving wire in field B

produces current I if there is a

conduction path

Induced emf

ji

k

z2)y1(y,

z1)(y,

Equations Iv vj

B Bi

FB Qv B QvBj i QvBk

Equations IIWork done per unit charge by F B

in moving charges from z1 to z2

vBl

where l z2 z1No work is done in moving charges in

other sections of path ( ignore Hall effect )

Work done per unit charge dW

dQ= emf

Thus

vBl

Equations IIIArea of loop in magnetic field

A t y t y1

lTotal magnetic flux through loop

t Rate of change of magnetic flux

d dt

-Bdydtl Bvl

1ytyBt AB

Faraday s Law of Induction for N loops

}{ AB ddt

dN

dt

d

This defines an Induced Electric Field

by

Faradays Law of Electromagnetic Induction

The work done per unit charge by magnetic force

moving charge from z1 to z2

dW

dQ 1

QF B d s

z 1

z2

1

QF B d s

loop E ind d s

loop

thus

N d

dt E ind d s

loop

An induced EMF is a measure An induced EMF is a measure ofof

An induced Electric FieldAn induced Electric Field If charge is in this region and If charge is in this region and there there is a conduction path it is a conduction path it will feel a will feel a force from the force from the induced Electric induced Electric Field and flow Field and flow

An induced EMF is a measure An induced EMF is a measure ofof

An induced Electric FieldAn induced Electric Field If charge is in this region and If charge is in this region and there there is a conduction path it is a conduction path it will feel a will feel a force from the force from the induced Electric induced Electric Field and flow Field and flow

Induced Electric Field

Equations

Remember for a static electric field E stat

Vab Estat d s

a

b

and

Estat d s 0 as Estat is conservative

But for an induced electric field E ind

Eind d s 0

thus

Eind is not conservative

Magnetic Flux and Induced Electric Field

Changing Magnetic Flux produces an

Induced Electric Field

Mechanical Work to Electrical work I

I Fappl

v

l

B

Blv

Pulling at constant velocity v

k

ji

y

Mechanical Work to Electrical work II

wire l with current I flowing in it

moving in a magnetic field B

feels a force given by

F Il B

F IlB k i IlB j

This force opposes the applied force F appl

and must be equal and opposite if the

velocity is to remain constant

F Fappl IlB

Mechanical Work to Electrical work III

I Fappl

v

l

B

Blv

F

Mechanical Power to Electrical Power IPower produced in circuit

I IBvlwhich is the same as the power produced by

the applied force

F appl v Fapplv IBlv

Mechanical Power to Electrical Power II

I Fappl

v

l

B

Blv

Pulling at constant velocity v

F

Magnetic Field produced by Changing Current

Circulating current produces an induced magnetic field

I

Bind

That opposes the external magnetic field B

That produces the current

Current produced by Changing Magnetic Field

(a) Change of External Magnetic Field Produces Current

(b) Current Produces Induced Magnetic Field

Lenz's Law

Lenz’s Law

Polarity of is such that it opposes the change that

caused it

Direction of Eis such that it opposes the change that

caused it

Direction of induced current is such that

it opposes the change that caused it

Conservation of Energy

Conservation of Energy

AC GeneratorAC Generator

AC Potential

t ddt

ddt

BA

ddtBACos BA sin t

if rotational speed is constant

t BAsin t maxsin t

max

BA

d

dt

t

DC GeneratorDC Generator

Eddy Currents