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ELECTROMAGNETIC INDUCTION

In the year 1820, it was discovered by Orested that an electric current produces a magnetic field. Soon after this, efforts were made to observe the converse of the magnetic effects of current, i.e. to show that magnetic field may produce electric current. Michel Faraday in England in 1831 demonstrated that electric current can be produced by employing a changing magnetic field. This phenomenon is called as electromagnetic induction. Faraday’s Laws of Electromagnetic Induction Faraday summed up his experimental results in the form of two laws known as Faraday’s Laws of electromagnetic induction .These are stated as follows First law: When the magnetic flux linked with the coil changes, an emf is induced in 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 rate of change of magnetic flux. Mathematically, d e dt d i.e. e  K dt Where K is constant of proportionality and is taken as 1. d  Induced emf e  dt The direction or sense of polarity of the induced emf is such that it tends to produce an induced current that will create a magnetic flux to oppose the change in the magnetic flux through the coil. This is known as Lenz’s Law and is stated below. Lenz’s Law Whenever an induced emf is set-up, the direction of the induced current through the loop is such that it opposes the cause which produces it. d Thus induced emf in a coil becomes e   dt The Lenz’s law is the consequence of the law of conservation of energy.

Integral and Differential form of Faraday’s Law of em induction Consider a closed circuit or a coil of any shape and is moving in a stationary r magnetic. Let S be the surface enclosed by the coil C. Let B  magnetic flux density in the neighborhood of the coil C. Then the magnetic flux through a small r r r elementary area dS is a scalar product B.dS .  Total magnetic flux through the entire coil is r r  B   B.dS S

According to Faraday’s law of electromagnetic induction the induced emf in a circuit is the –ve time rate of change of magnetic flux linked with the circuit. dB  induced emf e  dt d r r e    B.dS ----------------(1) dt S

Also by definition, the line integral of the electric field over a closed path give the r r e Ñ induced emf in the circuit --------------- E.dl

(2) r r Where E is the induced electric field at the current element dl of the closed circuit. From eq n s (1) and (2) r r d r r -----------------(3) Ñ  E.dl   dt S B.dS This eq n (3) is known as integral form of Faraday’s law of electromagnetic induction. Differential form : r If the circuit (coil C) remains stationary and only magnetic flux density B is changing then time derivative in eq n (3) may be taken inside the integral sign where it becomes a partial derivative. r r r B r i.e. --------------------(4) Ñ  E.dl   S t .dS Now, by Stoke’s Theorem r r r r E . dl  curl E .dS Ñ   S

 eq (5) becomes n

r r r B r S curl E.dS   S t .dS

Since the surface is arbitrary above eq n is true for any surface r r B  curl E   t r r B or ---------------(5)  E   t This is the differential form of Faraday’s law of electromagnetic induction. r r e Ñ E  .dl

Proof :

Consider a wire loop or frame of any shape which occupies the positions C1 at r time t. It is moving with a velocity v so that it occupies the position C2 at time t  dt . r Let elementary length dl of the loop is displaced r through a distance v .dt in the time dt , then the area r r dS swept by the element dl is given by r r r ----------------(1) dS  v .dt  dl r If B is the magnetic flux density t any point on this r r r area, then the magnetic flux the area dS is B.dS .Hence the total magnetic flux crossing the ribbon shaped surface S spanned by the boundary of the loop is r r   B.dS S

r r B The integral  .dS , therefore represent the change in magnetic flux crossing the S

wire loop, as it moves from position C1 to C2 in a time dt . r r d   B .dS  Thus S r Substituting the value of dS from eq n (1), we get r r r d    B.(v .dt  dl ) S

Now dt is independent of integration r r r d   B.(v  dl )  ---------------(2) dt S r r r r r r Now [ since cross and dot product are B.(v  dl )  ( B  v ).dl interchangeable ] r r r   (v  B ) . dl d r r r   Ñ (v  B) . dl -----------------(3)  dt r As the integration is now with respect to dl which is a line element and the integration is to be carried out over the boundary of the loop, the surface integral



has been changed to the line integral Ñ  . r r If E is the electric field associated with the elementary length dl when it is r moving with velocity v then r r r E  vB r Substituting the value of E in eq n (3) r r d   Ñ E . dl  dt d According to Faraday’s law , induced emf is e   dt r r e Ñ E . dl  r Thus induced emf = line integral of E over the circuit. S

SELF INDUCTION and COEFFICIENT of SELF INDUCTION The phenomenon due to which a coil opposes any change in the current that flows through it by inducing an opposing emf in itself is called as self induction. The induced emf is called as back emf and obeys the faraday’s law of electromagnetic induction. According to Lenz’s 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   LI or -----------(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

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  n eq (1) may be put as  L  I L   i.e. when I  1 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 faraday’s law induced emf in a coil is d d (L I ) d I e     L dt dt dt e L   dI dt dI  unity i.e. 1 A / s If then L   e dt 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 INDUCTANCE The 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 I1 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 the coil P. S  I p i.e. S  M I p

-----------(1) Where M is the constant of proportionality and is known as the coefficient of mutual induction or mutual inductance of coil S with respect to coil P. Now According to Faradays law of electromagnetic induction emf induced in the coil S to change in Current in the coil P is d  M Ip  d S eS   i.e. eS   dt dt eS   M

d  Ip  dt

or

M  

eS dI p

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

dt dI p

M  eS then  1A / s dt  the coefficient of mutual induction or mutual inductance of two coil is numerically equal to the emf induced in the secondary coil when the current flowing through the primary coil decreases at the rate of 1 A/s. S from eq n (1) M  Ip



i.e. M   S when I p  1 unit Thus coefficient of mutual inductance is numerically equal to the magnetic flux linked with the secondary coil when a unit current flows through primary coil. 1 2 2 2 Unit of M is henry denoted by H. Its dimensions are  M L T A  . ################################################## ####################