Chapter 20: Induced Voltages and Inductances

Induced Emf and Magnetic FluxHomework assignment : 17,18,57,25,34,66

Discovery of induction

Magnetic fluxA (area)

nAA ˆ B

magnetic flux: BAB

ABABBAB

cosmagnetic flux:

plane thelar toperpendicuector Unit v

runit vecto Normal : n̂

nAA ˆ

B

Induced Emf and Magnetic Flux

Farady’s law of induction

An emf in volts is induced in a circuit that is equal to the time rate of change of the total magnetic flux in webers threading (linking) the circuit:

The flux through the circuit may be changed in several different ways

1) B may be made more intense.

2) The coil may be enlarged.

3) The coil may be moved into a region of stronger field.

4) The angle between the plane of the coil and B may change.

Farady’s Law of Induction

tB

t

N B If the circuit containsN tightly wound loops

Farady’s law of induction (cont’d)

sd

Farady’s Law of Induction

•The sum of Ei si along the loopis equal to the work done per unitcharge, which is the emf of the circuit.

i

iiii

ii sEsE cos

Lenz’s law

Farady’s Law of Induction

The sign of the induced emf is such that it tries to produce a current that would create a magnetic flux to cancel (oppose) the original flux change.

B due to induced current

B due to induced current

Lenz’s law (cont’d)

• The bar magnet moves towards loop.

• The flux through loop increases, and an emf induced in the loop

produces current in the direction shown.

• B field due to induced current in the loop (indicated by the dashed

lines) produces a flux opposing the increasing flux through the loop

due to the motion of the magnet.

Farady’s Law of Induction

Motional Electromotive Force Origin of motional electromotive force I

FB

FE

Motional Electromotive Force Origin of motional electromotive force I (cont’d)

Motional Electromotive Force Origin of motional electromotive force II

B.Bind

Motional Electromotive Force Origin of motional electromotive force II (cont’d)

tB

Motional Electromotive Force Origin of motional electromotive force II (cont’d)

Blvt

xBl

t

xBl

tB

Motional Electromotive Force Origin of motional electromotive force II (cont’d)

Motional Electromotive Force Origin of motional electromotive force II (cont’d)

Motional Electromotive Force Origin of motional electromotive force III

Motional Electromotive Force Origin of motional electromotive force III (cont’d)

t

vmmaF t

0lim

dt

dvm

t

vm

R

vBt

0

22

lim

Motional Electromotive Force Origin of motional electromotive force III (cont’d)

dt

dvm

t

vm

R

vBt

0

22

lim

JUST FOR FUN WITH CALCULUS!

Motional Electromotive Force

A bar magnet and a loop (again)

In this example, a magnet is being pushed towards (away from) a closed loop.

The number of field lines linking the loop is evidently increasing (decreasing).

Motional Electromotive Force

An electromagnet and a coil

Motional Electromotive Force

Tape recorder

Generators

A generator (alternator)The armature of the generator opposite is rotating in a uniform B field with angular velocity ω this can be treated as a simple case of the E = υ×B field.

On the ends of the loop υ×B is perpendicular to the conductor so does not contribute to the emf. On the top υ×B is parallel to the conductor and has the value E = υB sin θ = ωRB sin ωt. The bottom conductor has the same value of E in the opposite direction but the same sense of circulation.

top

bottom

vv

B

i

iiAB tABtLRBsE sinsin2

Farady emf

Generators

A generator (cont’d)

AC generator

DC generator

Self-inductance

Self-inductance

Consider the loop at the right.

- Switch closed : Current starts to flow on the loop.- Magnetic field produced in the area enclosed by the loop (B proportional to I).- Flux through the loop increases with I.- Emf induced to oppose the initial direction of the current flow.- Self-induction: changing the current through the loop inducing an opposing emf the loop.

X X X X X X X

switch

X X X X X X X

a b

inducedcurrent

Self-inductance Self-inductance (cont’d) I

- The magnetic field induced by the current in the loop is proportional to the current:

- The magnetic flux induced by the current in the loop is also proportional to the current:

IB

i

iiB IsB

- Define the constant of proportion as L:I

L B self-inductance

- From Frarady’s law:t

IL

tB

L

2Wb T m1 H 1 1

A A SI unit of L :

Self Inductance Calculation of self inductance : A solenoid Accurate calculations of L are generally difficult. Often the answer depends even on the thickness of the wire, since B becomes strong close to a wire.

In the important case of the solenoid, the first approximation result for L is quite easy to obtain: earlier we had

Hence

Then,

IN

B0 I

ANNABB

2

0

AnAN

IL B 2

0

2

0

lengthunit per

turnsofnumber the: n

So L is proportional to n2 and the volume of the solenoid

Self Inductance Calculation of self inductance: A solenoid (cont’d)

Example: the L of a solenoid of length 10 cm, area 5 cm2, with a total of 100 turns is

L = 6.28×10−5 H

0.5 mm diameter wire would achieve 100 turns in a single layer.

Going to 10 layers would increase L by a factor of 100. Adding an iron or ferrite core would also increase L by about a factor of 100. 

The expression for L shows that μ0 has units H/m, c.f, Tm/A obtained earlier

lengthunit per

turnsofnumber the: n

An

ANL 2

0

2

0

RL Circuits Inductor

A circuit element that has a large inductance, such as a closely wrapped coil of many turns, is called a inductor.

RL circuitKirchhoff’s rules:

0 LIR

dt

dIL

t

IL LL

0dt

dILIR

)1()1( // tLRt eR

eR

I time constant

Energy Stored in a Magnetic Field Inductor

• The emf induced by an inductor prevents a battery from establishing instantaneous current in a circuit.

• The battery has to do work to produce a current – this work can be considered as energy stored in the inductor in its magnetic field.

2

2

1LIPEL energy stored in inductor