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Chapter 20: Induced Voltages and Inductances. Homework assignment : 17,18,57,25,34,66 . Discovery of induction. Induced Emf and Magnetic Flux. Induced Emf and Magnetic Flux. Magnetic flux. A (area). magnetic flux: . magnetic flux: . q. Farady’s law of induction.

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Chapter 20 induced voltages and inductances l.jpg

Chapter 20: Induced Voltages and Inductances

Homework assignment : 17,18,57,25,34,66

  • Discovery of induction

Induced Emf and Magnetic Flux


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Induced Emf and Magnetic Flux

  • Magnetic flux

A (area)

magnetic flux:

magnetic flux:

q


Farady s law of induction l.jpg

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:

If the circuit contains

N tightly wound loops

Farady’s Law of Induction

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

  • B may be made more intense.

  • The coil may be enlarged.

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

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


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Farady’s Law of Induction

  • The sum of Ei Dsi along the loop

  • is equal to the work done per unit

  • charge, which is the emf of the circuit.


Farady s law of induction5 l.jpg

B due to induced current

B due to induced current

  • Lenz’s law

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.

Farady’s Law of Induction


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Farady’s Law of Induction

  • 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.


Motional electromotive force l.jpg

FE

FB

Motional Electromotive Force



Motional electromotive force9 l.jpg

B

.

Bind

Motional Electromotive Force






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Motional Electromotive Force



Motional electromotive force16 l.jpg

JUST FOR FUN WITH CALCULUS!

Motional Electromotive Force


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Motional Electromotive Force

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 force18 l.jpg

Motional Electromotive Force


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Motional Electromotive Force


Generators l.jpg

v

v

  • A generator (alternator)

q

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

90o-q

B

Generators

bottom

Farady emf


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AC generator

Generators

DC generator


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induced

current

a

b

X X

X X X

X X

X X

X X X

X X

  • Self-inductance

Consider the loop at the right.

switch

  • - 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.

Self-inductance


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I

  • Self-inductance (cont’d)

- 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:

self-inductance

- Define the constant of proportion as L:

- From Frarady’s law:

Self-inductance

SI unit of L :


Slide24 l.jpg

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,

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


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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


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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 circuit

Kirchhoff’s rules:

time constant


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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.

energy stored in inductor


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