# Electromagnetic Induction

#### Magnetic flux

The total number of magnetic lines of force passing through a given area normally is called magnetic flux.

If an elementary area $\delta \overrightarrow{A}$ is placed in a magnetic field such that its normal makes an angle $\theta$ with direction of field lines. Then magnetic flux linked through area:

$\delta \emptyset = B \delta A cos \theta = \overrightarrow{B} \overrightarrow{ \delta A}$

Magnetic Flux

Therefore, magnetic flux linked through whole surface area $\emptyset = \sum \overrightarrow{B} \overrightarrow{ \delta A}$

The unit of magnetic flux is weber.

#### Electromagnetic Induction

Whenever there is change in magnetic flux linked with a circuit, an e.m.f is induced in the circuit. If the circuit is closed, a current is also induced in it.

The E.M.F and current so produced last so long as the change in flux linked with the circuit lasts. The phenomenon is called electromagnetic induction.

#### Faraday’s law of Electromagnetic Induction

The expression observations of electromagnetic induction have been summed up by Faraday in the form of two laws:

i. Whenever there is a change in magnetic flux linked with circuit, an e.m.f is set up in the circuit. The magnitude of induced e.m.f is proportional to the rate of the change of magnetic flux linked with the circuit. If $\emptyset$ is magnetic flux linked with the circuit at any instant t, then induced e.m.f.

$e \propto \dfrac{d \emptyset}{d t} \cdots equation \, \, 1$

ii. The direction of induced e.m.f is such that it opposes the change in magnetic flux that produces it. This law is also called Lenz’s law.

In view of second law, equation (1) takes the from $e \propto - \dfrac{d \emptyset}{dt} \, \, \, or \, \, \, e = - K \, \dfrac{d \emptyset}{dt}$

In S.I. system constant K = 1.

$e = - \dfrac{d \emptyset}{dt} \cdots equation \, \, 2$

Conventionally the change in flux is given with one turn, if coil contains N-turns, the expression (2) modifies to $e = - N \dfrac{d \emptyset}{dt} \cdots equation \, \, 3$

If R is resistance of circuit, then current induced $I = \dfrac{e}{R} = - \dfrac{N}{R} \dfrac{d \emptyset}{dt} \cdots equation \, \, 4$

The charge induced in time dt is given by:

$dq = I d t = - \dfrac{N}{R} \dfrac{d \emptyset}{dt} \, \, = - \dfrac{N d \emptyset}{R}$

But $N \emptyset = effective \, \, \, flux \, \, \, linkage \, \, \, = \emptyset$

$dq = \dfrac{d \emptyset}{R} = \dfrac{Net \, \, change \, \, in \, \, flux}{resistance} \cdots equation \, \, 5$

Obviously the charge induced is independent of time.

#### Lenz’s Law

The Lenz’s law is based on conservation of energy and it gives the direction of induced e.m.f or current in the coil. The Lenz’s law simply states that current induced opposes the cause producing it. When north pole of a magnet is moved towards the coil, the induced current flows in a direction so as to oppose the motion of the magnet towards the coil.

This is only possible when nearer face of the coil acts as a magnetic north pole which necessitates an anticlockwise current in the coil. Then the repulsion between two similar poles opposes the motion of the magnet towards the coil.

Lenz's Law

Similarly, when the magnet is moved away from the coil, the direction of induced current in such as to make the nearer face of the coil as a south pole which necessitates a clockwise induced current in the coil. Then the attraction between two opposite poles opposes the motion of the magnet away from the coil.

In either case, therefore work has to be done in moving the magnet. It is this mechanical work, which appears as electrical energy in the coil. Thus the proportional of induced current in the coil is in accordance with the law of conservation of energy.

#### Induced e.m.f in a conducting rod moving through a uniform magnetic field

Let a thin conducting rod ab of length l move in a uniform magnetic field B directed perpendicular to plane of paper, downwards. Let the velocity v of the rod be in the plane of paper towards right. By Fleming left hand rule a free rlectron will experience a force evB directed from a to b along the length of the rod.

Uniform magnetic field

Due to this force the free electrons of rod move from ‘a’ to ‘b’ thus making end ‘b’ negative and end ‘a’ positive. This causes a potential difference along the ends of rod. This is induced e.m.f.

If E is electric field developed in the rod, then $E = \dfrac{V}{I}$

V being e.m.f induced across the rod. The electric field produced exerts a force eE on the electron upward i.e. opposite to direction of magnetic force.

In equilibrium of charges:

Electrical force = magnetic force

$e E = e v B \, \, \, or \, \, \, E = v B$

Therefore,

Induced e.m.f V= El = Bvl.

If the rod moves across the magnetic field making an angle $\theta$ with it, then induced e.m.f.

$V = B_n v l$

Where $B_n$ is component of magnetic field normal to $\overrightarrow{v}$ . here $B_n = B sin \theta$ .

Therefore, Induced e.m.f E = bvl sin $\theta$

The direction of induced current is given by Fleming Right Hand Rule which states  that if the forefinger, middle finger and thumb of right hand are arranged mutually perpendicular in such a way that forefinger points along the magnetic field, the thumb along the direction of motion of conductor, then the middle finger points along the direction of induced current.

#### Self-Inductance

Whenever the electric current flowing through a circuit changes, the magnetic flux flowing through a circuit changes, the magnetic flux linked with the circuit also changes. As a result an induced e.m.f is set up in the circuit. This phenomenon is called self-induction and the induced e.m.f is called the back e.m.f.

(i) If ‘I’ is the current flowing in the circuit, the flux linked with the circuit:

$\emptyset \propto i$ or $\emptyset = L I \cdots equation \, \, 1$

Where ‘L’ is called the self-inductance of the coil and its unit is henry.

From equation (1) L = $- \dfrac{\emptyset}{i}$

If I = 1 and l =$\emptyset$ .

Thus the self inductance of a circuit is numerically equal to the e.m.f induced in the circuit when the rate of change of current in the circuit is 1 amp/sec.

(iii) Also work done against back e.m.f in time dt, when back e.m.f is e and current in circuit is I, is given by :

$dW = - e dt = L \dfrac{di}{dt} dt = L idi$

The total work done in establishing the current from 0 to I is given by:

This work is stored as the energy of the magnetic field. From expressions (1), (2) and (3), we have three alternative definitions of self-inductance.

From (2), L = $- \dfrac{e}{\dfrac{di}{dt}}$

From (3), L =$\dfrac{2W}{i^2}$

Thus the self-inductance of a circuit is numerically equal to twice the work done against the induced e.m.f in establishing a current of ‘I’ amp in the coil.

Role of Self Inductance:

The role of self-inductances in an electrical circuit is the same as that of the inertial in mechanical motion. Thus the self-inductions of a coil is a measure of ability to oppose the changes in current through it.

#### Self-inductance of a Solenoid

Consider a solenoid of length ‘l’ and number of turns ‘N’ and cross sectional area ‘A’. Then the number of turns per unit length,

$n = \dfrac{N}{l}$

Self Inductance

Magnetic field within air-solenoid, $B = \mu _0 ni$

$= \mu _0 \dfrac{N}{l} i$

The magnetic flux linked with the solenoid $\emptyset = NBA$

$= N ( \dfrac{\mu _0 Ni}{i} ) A = \dfrac{\mu _0 N^2 A}{l} i$

According to definition of self-inductance,

$L = \dfrac{\emptyset}{i} = \dfrac{ ( \mu _0 N^2 A I / l ) }{i} = \dfrac{\mu _0 N^2 A}{l}$

If solenoid is wound over a core of permeability $\mu$ then,

$Self \, \, Inductance \, \, L = \dfrac{\mu N^2 A}{l} = \dfrac{\mu _r \mu _0 N^2 A}{l}$

#### Inductance in series and parallel connection

If inductance $L_1 \, \, and \, \, L_2$ are connected in series than,

$net \, \, inductance \, \, L = L_1 + l_2$

If inductances $L_1 \, \, and \, \, L_2$ are connected in parallel, then net inductance is given by:

$\dfrac{1}{L} = \dfrac{1}{L_1} + \dfrac{1}{L_2} \rightarrow \dfrac{L_1 L_2}{ ( L_1 + L_2 )}$

Magnetic energy stored in inductance: $U_m = \dfrac{1}{2} Li^{-2} = \dfrac{B^2}{2 \mu} \times volume$

#### Mutual Inductance

Consider two coils $C_1 \, \, and \, \, c_2$ placed near each other such that if a current passes in coil $c_1 and C_2$ is in the magnetic field of coil $C_1$ and vice versa.

Whenever the current flowing through a coil $( C_1 )$ changes, the magnetic flux linked with the neighboring coil $( C_2 )$ also changes. This causes an induced e.m.f and hence an induced current in coil $C_2$ . This phenomenon is called mutual induction. The circuit in which the current changes is called the primary circuit, while the neighboring circuit in which e.m.f is induced is called the secondary circuit.

(i) If $i_1$ is current flowing through primary coil at any instant, the flux linked with secondary coil is given by:

$\emptyset _2 \propto i_1 \, \, \, or \, \, \, \emptyset _2 = Mi_1 \cdots equation \, \, \, 1$

Where ‘M’ is called the mutual inductance of the coils.

From (1), $M = \dfrac{\emptyset _2}{i_1}$

If $i_1 = 1 \, \, amp \, \, \, M = \emptyset _2$

Thus the mutual inductance of two neighboring coils is defined as the flux linked with a coil when the current in the neighboring coils is 1 amp.

(ii) Also induced e.m.f. in secondary coil,

$e_2 = - \dfrac{d \emptyset _2}{dt} = - \dfrac{d}{dt} ( Mi_1 ) = - M \dfrac{di_1}{dt} \cdots equation \, \, \, 2$

From (2), $M = - \dfrac{e_2}{ ( \dfrac{di_1}{dt} )}$

Thus the mutual inductance of two coils is defined as the e.m.f. induced in a coil when the rate of change of current flowing in the neighboring coil is 1 amp/sec.

Like self-inductance, the unit of mutual inductance is henry. The direction of induced e.m.f or induced current arising due to a change in magnetic flux in all cases is given by Lenz’s law.

#### Eddy Currents

When the current in a circuit changes, the currents are induced in nearby conductor. These currents heat up the conductor and are called eddy currents.

The eddy currents may be reduced by using laminated iron cores.

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