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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

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

- 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

- 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 (cont’d)

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 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)

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).

Motional Electromotive Force

Motional Electromotive Force

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

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

- 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 :

- 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

- 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

- 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

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