Chapter 20: Induced Voltages and Inductances

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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

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

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.

Farady’s law of induction (cont’d)

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.

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

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.

Origin of motional electromotive force I

FE

FB

Motional Electromotive Force

Origin of motional electromotive force II

B

.

Bind

Motional Electromotive Force

Origin of motional electromotive force III (cont’d)

JUST FOR FUN WITH CALCULUS!

Motional Electromotive Force

A bar magnet and a loop (again)

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).

An electromagnet and a coil

Motional Electromotive Force

Tape recorder

Motional Electromotive Force

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

A generator (cont’d)

AC generator

Generators

DC generator

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

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 :

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

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

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

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