Electro-Magnetism & Electromagnetic Induction

Magnetostatics:

Absolute and Relative Permeabilities of a Medium:

Every medium is supposed to possess two permeabilities : (i) absolute permeability and (ii) relative permeability
For measuring relative permeability, vacuum or free space is chosen as the reference medium.
The unit is henry/metre.
It is allotted an absolute permeability of
Obviously, relative permeability of vacuum with reference to itself is unity.
Absolute permeability of any medium is product of absolute permeability of vacuum and relative permeability of the particular medium.

Laws of Magnetic Force:

Coulomb was the first to determine experimentally the quantitative expression for the magnetic force between two isolated point poles.
He found that the force between two magnetic poles placed in a medium is: (i) directly proportional to their pole strengths (ii) inversely proportional to the square of the distance between them and (iii) inversely proportional to the absolute permeability of the surrounding medium.

Magnetic Field Strength (H):

Magnetic field strength at any point within a magnetic field is numerically equally to the force experienced by a N pole of one weber placed at that point.
Hence, unit of H is N/Wb.
If a pole of m Wb is placed in a uniform field of strength H N/Wb, then force experienced by the pole is = mH newtons.
It should be noted that field strength is a vector quantity having both magnitude and direction.

Magnetic Potential:

The magnetic potential at any point within a magnetic field is measured by the work done in shifting a N pole of one weber from infinity to that point against the force of the magnetic field.
It is a scalar quantity.

Flux per Unit Pole:

A unit N pole is supposed to radiate out a flux of one weber.
Therefore, the flux coming out of a N pole of m weber is given by

Flux Density (B):

It is given by the flux passing per unit area through a plane at right angles to the flux.
It is usually designated by the capital letter B and is measured in weber/meter square
It is a Vector Quantity.
Relative permeability of a material is equal to the ratio of the flux density produced in that material to the flux density produced in vacuum by the same magnetizing force.

Intensity of Magnetization (I):

It may be defined as the induced pole strength developed per unit area of the bar.
Also, it is the magnetic moment developed per unit volume of the bar.

Susceptibility (K):

Susceptibility is defined as the ratio of intensity of magnetization I to the magnetizing force H.
K = I/H henry/metre

ELECTROMAGNETISM:

Force on a Current carrying Conductor Lying in a Magnetic Field:

Whenever a current carrying conductor is placed in magnetic field, it experiences a force which acts in a direction perpendicular both to the direction of the current and the field.
In the given figure a conductor XY lying at right angles to the uniform horizontal field of flux density B Wb/meter square, produced by two solenoids A and B.

If l is the length of the conductor lying within this field and I ampere is the current carried by it, then the magnitude of the force experienced by it can be given as F = BIl.
The direction of this force may be easily found by Fleming left hand rule.

Fleming left hand rule:

Hold out your left hand with forefinger, second finger and thumb at right angles to one another. If the forefinger represents the direction of the field and
the second finger that of the current, then thumb gives the direction of the motion.

Ampere Circuital Law:

The law states that m.m.f. (magnetomotive force corresponding to e.m.f. i.e. electromotive force of electric field) around a closed path is equal to the current enclosed by the path.
magnetomotive force corresponds to e.m.f. which is electromotive force of electric field.

Electromagnetism:

When a magnetic field embracing a conductor moves relative to the conductor, it produces a flow of electrons in the conductor.
This phenomenon produces an e.m.f. and hence current (i.e. flow of electrons) is induced in any conductor which is cut across or is cut by a magnetic flux is known as electromagnetic induction.

Faradays Laws of Electromagnetic Induction:

First Law states that:
Whenever the magnetic flux linked with a circuit changes, an e.m.f. is always induced in it.
or
Whenever a conductor cuts magnetic flux, an e.m.f. is induced in that conductor.
Second Law states that:
The magnitude of the induced e.m.f. is equal to the rate of change of flux-linkages.

Direction of induced e.m.f. and currents:

Fleming rule is used to find direction of e.m.f. where induced e.m.f. is due to flux-cutting (i.e., dynamically induced e.m.f.).
Lenz law is used to find direction of e.m.f. when e.m.f. is induced by change in flux-linkages (i.e., statically induced e.m.f.).

Fleming Right Hand Rule:

Here, the front side of the hand is held perpendicular to the incident flux with the thumb pointing in the direction of the motion of the conductor.
The direction of the fingers give the direction of the induced e.m.f. and current.

Lenz’s Law:

The electromagnetically induced current always flows in such direction that the action of the magnetic field set up by it tends to oppose the very cause which produces it.
When N-pole of the bar magnet approaches the coil, the induced current set up by induced e.m.f. flows in the anticlockwise direction in the coil as seen from the magnet side. The result is that face of the coil becomes a N-pole and so tends to oppose the onward approach of the N-Pole of the magnet (like
poles repel each other)
When the induced current flows in the clockwise direction thus making the face of the coil (facing the magnet) a S pole. Therefore, the N pole of the magnet has to withdrawn against this attractive force of the S-pole of coil.

Induced e.m.f.:

Induced e.m.f. can be either (i) dynamically induced or (ii) statically induced.
In the first case, usually the field is stationary and conductors cut across it (as in d.c. generators).
But in the second case, usually the conductors or the coil remains stationary and flux linked with it is changed by simply increasing or decreasing the current producing this flux (as in transformers).

Dynamically induced e.m.f:

When conductor or coil rotates in magnetic field the emf is induced dynamically.

Statically Induced E.M.F.:

In this the conductors or the coil remains stationary and flux linked with it is changed by simply increasing or decreasing the current producing this flux (as in transformers).
It can be further sub-divided into (a) mutually induced e.m.f. and (b) self induced e.m.f
The emf induced in a coil due to the change of flux produced by another neighboring coil linking to it, is called Mutually Induced emf.
Self-induced e.m.f. is the e.m.f. induced in a coil due to the change of its own flux linked with it.

Mutually Induced e.m.f.

The emf induced in a coil due to the change of flux produced by another neighboring coil linking to it, is called Mutually Induced emf.
Coil B is having N2 number of turns and is placed near another coil A having N1 number of turns.
Coil A is joined to a battery, a switch and a variable resistance R whereas coil B is connected to a sensitive voltmeter V.
When current through A is established by closing the switch, its magnetic field is set up which partly links with or threads through the coil B.
As current through A is changed, the flux linked with B is also changed.
Hence, mutually induced e.m.f. is produced in B whose magnitude is given by Faraday Laws and direction by Lenz Law.

There is no movement of any conductor, the flux variations being brought about by variations in current strength only.
Such an e.m.f. induced in one coil by the influence of the other coil is called (statically but) mutually induced e.m.f.

Self induced e.m.f

This is the e.m.f. induced in a coil due to the change of its own flux linked with it.
If current through the coil is changed, then the flux linked with its own turns will also change, which will produce in it what is called self induced e.m.f.
The direction of this induced e.m.f. (as given by Lenz law) would be such as to oppose any change of flux which is the very cause of its production.
Hence, it is also known as the opposing or counter e.m.f. of self induction.

Self Inductance

This property of the coil due to which it opposes any increase or decrease or current of flux through it, is known as self inductance.
It is quantitatively measured in terms of coefficient of self induction L.
This property is analogous to inertia in a material body.
Initially it is difficult to set a heavy body into motion, but once in motion, it is equally difficult to stop it.
Similarly, in a coil having large self-induction, it is initially difficult to establish a current through it, but once established, it is equally difficult to withdraw it.
Hence, self-induction is sometimes analogously called electrical inertia or electromagnetic inertia.

Coefficient of Self induction (L):

The coefficient of self induction of a coil is defined as the weber turns per ampere in the coil.
Weber-turns means the product of flux in webers and the number of turns with which the flux is linked.
In other words, it is the flux linkages of the coil.
Hence a coil is said to have a self-inductance of one henry if a current of 1 ampere when flowing through it produced flux linkages of 1 Weber turn in it.

Mutual Inductance:

It is defined as the ability of one coil (or circuit) to produce an e.m.f. in a nearby coil by induction when the current in the first coil changes.
This action being reciprocal, the second coil can also induce an e.m.f. in the first when current in the second coil changes.
This ability of reciprocal induction is measured in terms of the coefficient of mutual induction M.
Let there be two magnetically-coupled coils having N1 and N2 turns respectively.
Coefficient of mutual inductance between the two coils is defined as the weber turns in one coil due to one ampere current in the other.
It is supposed that whole of this flux links with the turns of the second coil.
Hence, two coils are said to have a mutual inductance of 1 henry is one ampere current when flowing in one coil produces flux linkages of one Weber turn in the other.

Coefficient of Coupling:

Consider two magnetically coupled coils A and B having N1 and N2 turns respectively.
Their individual coefficients of self induction are:

The constant k is called the coefficient of coupling and may be defined as the ratio of mutual inductance actually present between the two coils to the maximum possible value.

Inductances in Series:

Let two coils be joined in series that their fluxes (or m.m.fs) are additive i.e., in the same direction.

If fluxes are in opposite direction.

M = coefficient of mutual inductance

L1 = coefficient of self-inductance of 1st coil

L2 = coefficient of self-inductance of 2nd coil

Inductance in Parallel:

Two inductances of values L1 and L2 henry are connected in parallel.
Let the coefficient of mutual inductance between the two be M.
Let i be the main supply current and i1 and i2 be the branch currents.