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ELECTROMAGNETIC INDUCTIONIn the year 1820, it was discovered by Orested that an electric current produces amagnetic field. Soon after this, efforts were made to observe the converse of themagnetic effects of current, i.e. to show that magnetic field may produce electriccurrent. Michel Faraday in England in 1831 demonstrated that electric current canbe produced by employing a changing magnetic field. This phenomenon is called aselectromagnetic induction.Faradays Laws of Electromagnetic InductionFaraday summed up his experimental results in the form of two laws known asFaradays Laws of electromagnetic induction .These are stated as followsFirst law: When the magnetic flux linked with the coil changes, an emf isinducedin it which lasts so long as the change of magnetic flux continues.

Thus condition for an emf to be induced in a coil is changing magnetic flux.Second law: The magnitude of the induced emf is directly proportional to the rateof change of magnetic flux. Mathematically,

de

dt

i.e. de Kdt

Where K is constant of proportionality and is taken as 1.

Induced emfd

edt

The direction or sense of polarity of the induced emf is such that it tends to producean induced current that will create a magnetic flux to oppose the change in themagnetic flux through the coil. This is known as Lenzs Law and is stated below.Lenzs LawWhenever an induced emf is set-up, the direction of the induced current through theloop is such that it opposes the cause which produces it.

Thus induced emf in a coil becomesd

e dt

The Lenzs law is the consequence of the law of conservation of energy.

Integral and Differential form of Faradays Law of em induction

Consider a closed circuit or a coil of any shape and is moving in a stationarymagnetic. Let S be the surface enclosed by the coil C. Let B magnetic flux densityin the neighborhood of the coil C. Then the magnetic flux through a smallelementary area dS is a scalar product .B dS.Total magnetic flux through the entire coil is

.B

S

B dS

According to Faradays law of electromagnetic induction the induced emf in acircuit is the ve time rate of change of magnetic flux linked with the circuit.

induced emf Bd

edt

.S

de B dS

dt

rr----------------(1)


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Also by definition, the line integral of the electric field over a closed path give the

induced emf in the circuit .e E dl ----------------(2)Where E is the induced electric field at the current element dl of the closedcircuit.

From neq s (1) and (2)

. .S

d E dl B dS

dt

r rr r

-----------------(3)

This neq (3) is known as integral form of Faradays law of electromagneticinduction.Differential form :If the circuit (coil C) remains stationary and only magnetic flux density B is

changing then time derivative in neq (3) may be taken inside the integral sign whereit becomes a partial derivative.

i.e. . .S

B

E dl dS t

r rr

--------------------(4)Now, by Stokes Theorem

. .S

E dl curl E dS

neq (5) becomes . .S S

Bcurl E dS dS

t

r rr

Since the surface is arbitrary above neq is true for any surface

Bcurl Et

r

or

B

E t

r

---------------(5)This is the differential form of Faradays law of electromagnetic induction.

Proof : .e E dl Consider a wire loop or frame of any shape which occupies the positions 1C at

time t. It is moving with a velocity v so that it occupies the position 2C at timet dt .Let elementary length dl of the loop is displacedthrough a distance .v dt in the time dt, then the areadSswept by the element dl is given by

.dS v dt dl r ----------------(1)IfB is the magnetic flux density t any point on thisarea, then the magnetic flux the area dS is .B dS.Hencethe total magnetic flux crossing the ribbon shapedsurface S spanned by the boundary of the loop is

.S

BdS


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The integral .S

B dS , therefore represent the change in magnetic flux crossing the

wire loop, as it moves from position 1 2C t o C in a time dt.

Thus .S

d B dS

Substituting the value of dS fromn

eq (1), we get.( . )

S

d B v dt dl r

Now dt is independent of integration

.( )S

d B v dl

dt

rr r---------------(2)

Now .( ) ( ). B v dl B v dl r r

[ since cross and dot product areinterchangeable ]

( ) .v B dl r

( ) .d

v B dl

dt

rrr -----------------(3)

As the integration is now with respect to dl which is a line element and theintegration is to be carried out over the boundary of the loop, the surface integral

S

has been changed to the line integral .If E is the electric field associated with the elementary length dl when it is

moving with velocity v then

E v B r

Substituting the value ofE inneq (3)

.d

E dldt

rr

According to Faradays law , induced emf isd

edt

.e E dl Thus induced emf = line integral ofE over the circuit.

SELF INDUCTION and COEFFICIENT of SELF INDUCTIONThe phenomenon due to which a coil opposes any change in the current that flowsthrough it by inducing an opposing emf in itself is called as self induction. Theinduced emf is called as back emf and obeys the faradays law of electromagneticinduction. According to Lenzs law this induced emf have a direction so as to oppose

the cause (changing current ) due to which it is produced .Coefficient of Self Induction or Self Inductance (L)Whenever a current is passed through a coil magnetic field is produced in the surrounding of the coil. The

number of lines of induction passing normally through an area near the coil i.e. magnetic flux is found to

be directly proportional to the current passing through the coil.I

or L I -----------(1)Where L is constant of proportionality and is called as coefficient of self induction or self

inductance of the coil. Its value depends upon the following factors


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1. The number of turns of the coil N.

2. Length of the coil

3. Area of cross-section of the coil A

4. Nature of the material of the core on which coil is wound neq (1) may be put as

LI

i.e. L when 1 I unit

Thus self inductance of a coil is numerically equal to the magnetic flux linked with the coil when

unit current flows through it. The SI unit of L is henry (H).Also according to faradays law induced emf in a coil is

de

dt

( )d L I

dt

d IL

dt

e

Ld I

dt

If . . 1 /d I unity i e A sdt

then L e

Thus self inductance of a coil is numerically equal to the induced emf when the current flowing

through it changes at the rate of unity (1 /A s ).

1 henry The self inductance of a coil is said to be 1 henry when a current changing at the rate of 1 A/s

through it induces an emf of 1 volt in it.NOTE: Inductance in a circuit plays the analogous role as mass in mechanics. Mass opposes the motion of a

particle and inductance opposes the change in the current. In other words the effect of inductance in a circuit is

same as inertia in mechanics and inductance is therefore called as electrical inertia.

MUTUAL INDUCTANCE and COEFFICIENT of MUTUAL INDUCTANCEThe phenomenon by virtue of which an induced emf is produced in a coil due to change in current in a

neighboring coil is called as mutual induction.Consider two coils P and S close to each other. Let 1I be the current flowing in the coil P at

some instant t and 2 be the magnetic flux linked with the coil S at that instant.Now flux linked with the coil S is directly proportional to the current flowing in thecoil P.i.e. S pI

S pM I -----------(1)Where M is the constant of proportionality and isknown as the coefficient of mutual induction ormutual inductance of coil S with respect to coil P.

Now According to Faradays law of electromagneticinduction emf induced in the coil S to change in

Current in the coil P is

SS

de

dt

i.e.

pS

d M Ie

dt

pS

d Ie M

dt or


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S

p

eM

dI

dt

------------------(2)

1 /pdI

A sdt

then SM e

the coefficient of mutual induction or mutual inductance of two coil isnumerically equal to the emf induced in the secondary coil when the current flowingthrough the primary coil decreases at the rate of 1 A/s.

from neq (1)S

p

MI

i.e. SM when 1p I unit

Thus coefficient of mutual inductance is numerically equal to the magnetic fluxlinked with the secondary coil when a unit current flows through primary coil.

Unit of M is henry denoted by H. Its dimensions are1 2 2 2M L T A .

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