electromagnetic induction part 1 · 2020-05-28 · electromagnetic induction -i wednesday, 27 may...

Electromagnetic Induction -I Wednesday, 27 May 2020

ELECTROMAGNETIC INDUCTION

THE PHENOMENON OF ELECTROMAGNETIC INDUCTION

An EMF is induced in a coil in a magnetic field whenever the flux (Φ) through the coil changes.

Some ways of achieving the flux change are shown below. (Note: The EMF ceases to exist once the change has taken place.)

The decreased current through the solenoid decreases the flux through the coil.

Increasing the separation of the coil and the magnet decreases the flux through the coil.

Turning the coil causes less flux to pass through it .

The effect is called electromagnetic induction and if the coil forms a part of a closed circuit, the induced EMF causes a current to flow in the circuit.

This effect was discovered by Faraday, and independently by Henry , in 1831, eleven years after Oersted’s discovery that a current-carrying conductor has an associated magnetic field.

1 Prabin Moktan, St. Augustine’s


Experiments show that the magnitude of the induced EMF depends on the rate of change of flux through the coil and the number of turns (N) in the coil as well.

We therefore define a term called flux-linkage as

flux-linkage = Nφ

It is not necessary for a conductor to be in the form of a coil in order for it to be able to acquire and induced EMF. Experiments show that EMF can be induced in the a straight conductor whenever it is caused to cut across magnetic field lines. The magnitude of the induced EMF is proportional to the rate of cutting of the flux.

Thus to summarise , an EMF is induced

when the flux through a coil changes

when a conductor cuts across magnetic field lines.

Though the aforementioned appear to be distinct ways of inducing EMF they are in fact equivalent.

THE LAWS OF ELECTROMAGNETIC INDUCTION

A detailed investigation of electromagnetic induction leads to two laws.

(i) The magnitude of the induced EMF in a circuit is directly proportional to the rate of change of flux-linkage or to the rate of cutting of magnetic flux. (Faraday’s Law).

(ii) The direction of the induced EMF is such that the current it causes to flow (or would flow in a closed circuit) opposes the change producing it (Lenz’s law).

The two laws can be expressed as Neumann’s equation:

where

E = induced EMF in volts

= the rate of change of flux linkage in Webers per second .

Notes

(i) If a non consistent set of units were being used , a constant of proportionality would be included in the above equation.

(ii) The minus sign takes into account Lenz’s law. According the Lenz’s law , the induced current flows in such a sense as to create a flux in the opposite direction to that in which the external flux has increased i.e. the current flows in such a direction as to oppose the change which has taken place.

E = −d(Nφ)

dt

d(Nφ)dt


Electromagnetic Induction -I Wednesday, 27 May 2020(iii) When working in terms of rate of cutting of flux rather than rate of change of

flux-linkage ,

where = the rate of cutting of flux

= the number of conductors cutting the flux.

AN ILLUSTRATION OF LENZ’S LAW

In the figure given below , the strength of the magnetic field at the solenoid increases as the magnet is moved towards it. An EMF is induced in the solenoid and the galvanometer indicates that a current is flowing. If a preliminary experiment has been performed to determine the direction of the current through the galvanometer which corresponds to a deflection in a particular sense , then it is seen that the current through the solenoid is in the direction that makes end A a North Pole. This opposes the motion of the magnet ( like poles repel) i.e. the direction of the current is such as to oppose the change which has induced it- Lenz’s Law.

The presence of North Pole at A means work has to be done to move the magnet and cause the current to flow. The work done is converted into electrical energy , some of which is dissipated as heat in the circuit and some of which provided the mechanical energy to deflect the galvanometer.

Lenz’s Law is the law of conservation of energy expressed in such a way as to apply specifically to electromagnet induction.

FLEMING’S RIGHT HAND RULE

The direction of an induced current can be always found by using Lenz’s Law, but if the current is being induced by the motion of a straight conductor , it is more convenient to use Fleming’s right hand (dynamo) rule :

If the first and second fingers and the thumb of the right hand are placed comfortably at right angles to each other , with the first finger pointing in the direction of the field and the thumb pointing in the direction of the motion , then the second finger points in the direction of the induced current.

E = − Ndφdt

dφdt

N


Electromagnetic Induction -I Wednesday, 27 May 2020EMF IN A STRAIGHT CONDUCTOR

The figure given below illustrates five distinct ways in which a straight conductor of length can be moved with velocity through a magnetic field of flux density . The only motion which induces an EMF between the ends of the conductor is that shown in (a) where the conductor cuts across the filed lines.

The EMF , , is given by

We are concerned only with the EMFs between the ends of the conductor , but it should not be overlooked that real conductors are of finite width, in which case the motions in (b) and (c) induce EMF across the width of the conductor.

Derivation of from Faraday’s Law

In figure (a) above ,

The area swept out by the conductor in 1 s =

The flux cut by the conductor in 1 s =

i.e The rate of cutting of flux =

In consistent units the relevant version of Faraday’s law can be written as

EMF = Rate of cutting of flux

L v B

E

E = BLv

E = BLv

Lv

BLv

BLv

EMF = BLv



Derivation of from Conservation of Energy

Suppose the conductor given below forms a part of a closed circuit and the induced EMF causes current to flow through it.

By Fleming’s right-hand rule the current is in the direction shown. Since the conductor is carrying a current and is in a magnetic field there will be a force acting on it. The magnitude of this force is given by

, and since , . By Fleming’s left-hand rule this force is towards the left, i.e. it opposes the applied force.

The conductor moves with constant velocity therefore the net force on it is zero, i.e.

Where is the magnitude of the applied force.

The applied force is moving its point of application and therefore is doing work (against . The rate at which this work is being done is given by ( )

Rate of working =

Rate of working =

This work is being converted to electrical energy at a rate (the electrical power). By conservation of electrical energy the rate of working is equal to the rate of production of electrical energy , i.e.

E = BLv

I

BILsinθ θ = 90o BIL

F = BIL

F

BIL) P = Fv

Fv

BILv

EI

EI = BILv

E = BLv



EMF INDUCED IN A RECTANGULAR COIL

Consider a rectangular coil PQRS of one turn situated in a magnetic field of flux density as shown below. The plane of the coil is perpendicular to the field and its dimensions are as shown.

Suppose the coil is moved sideways with velocity . As the coil moves to P’Q’R’S’ , both PS and QR cut across the field lines and equal and opposite EMFs are induced in them. The net EMF in the coil is therefore zero- there has been no change in flux through the coil. (There is no induced EMF in either PQ or SR because each of these

sections are moving parallel to its length. )

As the coil moves to P”Q”R”S” , QR leaves the field before PS. Once QR has left the field there is no EMF across it to oppose that across PS, and the coil as a whole has an induced EMF, , where

Derivation of on the Basis of Changing Flux-linkage

When all the coil (in the diagram above) is in the field the flux-linkage is ; when all of it is out of the field the flux-linkage is zero. Therefore ,

Change in flux-linkage =

The flux-linkage starts to change when QR leaves the field, and stops changing when PS leaves the field. The time taken for this is the time taken by the coil to travel a distance equal to its width, x.

Time taken forever flux-linkage to change =

B

v

E

E = BLv

E = BLv

BL x

BL x

∴xv



Rate of change of flux-linkage = ∴BL x

xv

= BLv ∴ E = BLv


electromagnetic induction part 1 · 2020-05-28 · electromagnetic induction -i wednesday, 27 may...

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