**Understandings**

➔ Electromotive force (emf)

➔ Magnetic flux and magnetic flux linkage

➔ Faraday’s law of induction

➔ Lenz’s law

**Applications ****and** **skills**

➔ Describing the production of an induced emf by a changing magnetic flux and within a uniform magnetic field

➔ Solving problems involving magnetic flux, magnetic flux linkage, and Faraday’s law

➔ Explaining Lenz’s law through the conservation of energy

### Equations

➔ Flux: Φ = *BA *cosθ

➔ Faraday’s / Neumann’s equation: ε = –*N* (dΦ/dt)

➔ emf induced in moving rod: ε = *Bvl*

➔ in side of coil with *N *turns: ε = *BvlN*

### 11.1 Electromagnetic Induction

#### Electromagnetic induction: When an electric charge moves in a magnetic field, then a force acts on it. In a reverse sense, a movement or change in magnetic field relative to stationary charge gives raise to an electric current.

**Induced emf (ε)**

Definition: Potential difference generated by electromagnetic induction.

For a rod of length L moved with velocity v in a region of magnetic field B:

If the rod moves from left to right, and thus, its electrons move perpendicular to the magnetic field, they experience a downward force along the rod and an electric field is established.

Flow of electrons quickly stops due to electrostatic repulsion at the bottom, and thus, the current exists for a short period of time.

Without movement, emf is not induced.

Formula if the rod is moved connected to wires (the work done to separate electrons leads to an induced emf): ε = BvL.

### Magnetic flux (Ф)

Definition: “Product of the magnitude of the normal component of magnetic field strength and area through which it passes.”

Intuitive picture: Number of magnetic field lines crossing a certain area.

Formula: Ф = BAcosθ, where A is the area and θ is the angle between the magnetic field strength direction and the direction normal to the loop area.

Units: weber (Wb)

Definition for a rod: “Product of magnitude and the rate at which the area swept out by the rod is changing” = ∆Ф/∆t.

Magnetic flux linkage: Magnetic flux multiplied by the N turns in a loop. Ф = NBAcosθ.

Magnetic flux density: numerically equivalent to magnetic field strength.

Induced emf = magnetic flux density x rate of change of area = B∆A/∆t.

### Faraday’s Law

Definition: “Induced emf is equal to the negative rate of change of magnetic flux linkage.”

Negative sign exists due to Lenz’s law (see below).

Formula: ε = -N∆Ф/∆t.

Magnetic field away from viewer

Coil of area A with N turns

Example of Faraday’s Law:

A coiled wire is moving into a magnetic field.

The induced current should create a magnetic field in a direction opposed to the existing field (in this case, opposing “away from”, therefore, towards the viewer)

Rod (perpendicular to field): in time ∆t, a rod of length L will move a distance s = v∆t, cutting magnetic field lines as it moves in the magnetic field. A = Ls

Formula: ∆Ф = ∆BAcos0º = ∆BA = ∆BLs = BLv∆t, and hence, ε = BvL.

### Lenz’s Law

Definition: “The induced emf will be in such a direction to oppose the change in the magnetic flux that crea

ted the current”. It is equivalent to energy conservation.

Work done by magnetic forces that arises due to current is dissipated as thermal energy.

Examples:

Rod: Force in the rod must oppose

Use left-hand rule twice: Firstly to find the direction of the current in the loop. Secondly, to find the force induced on the rod due to the current the motion. Hence, if it moves towards the right, a leftwards force will appear indicating a counter-clockwise induced current.

Loop wire and a wire with increasing current: Magnetic flux is increasing into the page. Hence, to oppose the increase in magnetic flux (inside the loop), a magnetic field out of the page must exist, and thus, a counter-clockwise current is induced.

Bar magnet through a loop of wire:

When approaching the loop, magnetic flux is increasing, and thus, magnetic field must oppose the increase, with a counter-clockwise current.

When leaving the loop, the magnetic flux is decreasing, and the current is now clockwise.

The opposite magnet (south pole first) would have the exact opposite effect.

### ELECTROMAGNETIC INDUCTION

### THE EXPERIMENTS OF FARADAY AND HENRY

### MAGNETIC FLUX3>
The number of magnetic lines of force crossing a surface is called magnetic flux linked with the surface.
It is represented by .
Magnetic flux
where B is strength of magnetic field, A is area of the surface and θ is the angle which normal to the area (unit area vector) makes with the direction of magnetic field.
The S.I. unit of magnetic flux is weber which is the amount of magnetic flux over an area of 1 m2 held normal to a uniform magnetic field of one tesla.
The c.g.s. unit of φ is maxwell.
1 weber = 108 maxwell.
### FARADAY’S LAW OF ELECTROMAGNETIC INDUCTION3>
Whenever the number of magnetic lines of force (flux) linked with any closed circuit change, an induced current flows through the circuit which lasts only so long as the change lasts. An increase in the number of lines of force produces an inverse current, while a decrease of such lines produces a direct current.
The induced emf is equal to the negative rate of change of magnetic flux.
i.e.
The -ve sign shows that the induced emf opposes the change in magnetic flux (Lenz’s law).
### LENZ’S LAW3>
The direction of induced e.m.f. is given by Lenz’s law. According to this law, the direction of induced e.m.f. in a circuit is always such that it opposes the every cause which produces it.
Thus,
Lenz’s law is in accordance with the principle of conservation of energy. Infact, work done in moving the magnet w.r.t. the coil changes into electric energy producing induced current.
There is also another law for finding the direction of induced current. This is Fleming’s right hand rule. According to this rule, if we stretch the right-hand thumb and two nearby fingers perpendicular to one another such that the first finger points in the direction of magnetic field and the thumb in the direction of motion of the conductor, then the middle finger will point in the direction of the induced current. Total flow of charge due to change of flux (Δφ):### METHODS OF INDUCING E.M.F.3>As is known, e.m.f. is induced in a circuit only when amount of magnetic flux linked with the circuit changes. As φ = BA cos θ, therefore three methods of producing induced e.m.f. :- By changing B
- By changing A
- By changing θ (orientation of the coil)

When a conductor of length l moves with a velocity v in a magnetic field of strength B so that magnetic flux linked with the circuit changes, the e.m.f. induced (ε) is given byε = B l v.#### INDUCED E.M.F. AND ITS DIRECTION4>Case (i) In conducting rodThe induced e.m.f. is generated because of rotation of a conducting rod in a perpendicular magnetic field also, e = – BAf where f = frequency of rotation and A = πr2, where r is the radius of circle in which this rod moves, hence r = l.

ω = angular velocity, l = length of conducting rod.

### LENZ’S LAW3>
The direction of induced e.m.f. is given by Lenz’s law. According to this law, the direction of induced e.m.f. in a circuit is always such that it opposes the every cause which produces it.
Thus,
Lenz’s law is in accordance with the principle of conservation of energy. Infact, work done in moving the magnet w.r.t. the coil changes into electric energy producing induced current.
There is also another law for finding the direction of induced current. This is Fleming’s right hand rule. According to this rule, if we stretch the right-hand thumb and two nearby fingers perpendicular to one another such that the first finger points in the direction of magnetic field and the thumb in the direction of motion of the conductor, then the middle finger will point in the direction of the induced current. Total flow of charge due to change of flux (Δφ):### METHODS OF INDUCING E.M.F.3>As is known, e.m.f. is induced in a circuit only when amount of magnetic flux linked with the circuit changes. As φ = BA cos θ, therefore three methods of producing induced e.m.f. :- By changing B
- By changing A
- By changing θ (orientation of the coil)

When a conductor of length l moves with a velocity v in a magnetic field of strength B so that magnetic flux linked with the circuit changes, the e.m.f. induced (ε) is given byε = B l v.#### INDUCED E.M.F. AND ITS DIRECTION4>Case (i) In conducting rodThe induced e.m.f. is generated because of rotation of a conducting rod in a perpendicular magnetic field also, e = – BAf where f = frequency of rotation and A = πr2, where r is the radius of circle in which this rod moves, hence r = l.

ω = angular velocity, l = length of conducting rod.

- By changing B
- By changing A
- By changing θ (orientation of the coil)

#### INDUCED E.M.F. AND ITS DIRECTION4>Case (i) In conducting rodThe induced e.m.f. is generated because of rotation of a conducting rod in a perpendicular magnetic field also, e = – BAf where f = frequency of rotation and A = πr2, where r is the radius of circle in which this rod moves, hence r = l.

ω = angular velocity, l = length of conducting rod.

ω = angular velocity, l = length of conducting rod.

ω = angular velocity of disc.

- if key K is closed then current in P will flow in clockwise direction and consequently induced current in Q will flow in anticlockwise direction. (see fig.a)
- when key K is opened then current in P falls from maximum to zero and consequently induced current in Q will flow in clockwise direction. (see fig.b)

- the direction of induced current in the loop will be clockwise so that it may oppose the increase of magnetic flux in the loop in downward direction.

- the direction of induced current in the loop will be anti-clockwise so that it may oppose the increase of magnetic flux in the loop in upward direction.

- Induced electric field lines form closed loops (different from the electric field lines used to depict electric field produced due to charges)
- Induced electric field is non-conservative in nature (again a difference from the electric field produced by electric charges)

- An emf is induced in a circuit where the magnetic flux is changing even if the circuit is open. But obviously no current will flow. If we close the circuit, the current will start flowing.
- In a loop moving in a uniform magnetic field, when the loop remains in the field, the net emf induced is zero.

### EDDY CURRENTS3>The induced circulating currents produced in a metal itself due to change in magnetic flux linked with the metal are called eddy currents. These currents were discovered by Foucault, so they are also known as Foucault Currents.The direction of eddy currents is given by Lenz’s law.Eddy currents produced in a metallic block moving in a non-uniform magnetic field is shown in fig.#### APPLICATIONS OF EDDY CURRENT4>Like friction, eddy currents are helpful in some fields and have to be increased, while in some other fields they are undesirable and have to be minimised.- Dead beat galvanometer
- Energy meter
- Speedometer
- Electric brakes
- Single phase AC motor
- Induction furnace
- Diathermy

- Dead beat galvanometer
- Energy meter
- Speedometer
- Electric brakes
- Single phase AC motor
- Induction furnace
- Diathermy

### SELF INDUCTANCE AND MUTUAL INDUCTANCE3>#### SELF INDUCTANCE4>The property of a coil by virtue of which the coil opposes any change in the strength of the current flowing through it, by inducing an e.m.f. in itself is called self inductance.When a current I flows through a coil, the magnetic flux φ linked with the coil is φ = LI, where L is coefficient of self inductance of the coil.On differentiating, we getIf dI / dt = 1; L = – e. Hence coefficient of self inductance of a coil is equal to e.m.f. induced in the coil when rate of change of current through the same coil is unity. Coefficient of self induction of a coil is also defined as the magnetic flux linked with a coil when 1 ampere current flows through the same coil.The value of L depends on geometry of the coil and is given bywhere l is length of the coil (solenoid), N is total number of turns of solenoid and A is area of cross section of the solenoid.The S.I. unit of L is henry. Coefficient of self induction of a coil is said to be one henry when a current change at the rate of 1 ampere/sec. in the coil induces an e.m.f. of one volt in the coil.

- Energy stored in a coil (inductor) =

- The self inductance is a measure of the coil to oppose the flow of current through it. The role of self-inductance in an electrical circuit is the same as that of the inertia in mechanics. Therefore it is called electrical inertia.
- The magnetic energy density (energy stored per unit volume) in a solenoid

#### MUTUAL INDUCTANCE4>Mutual induction is the property of two coils by virtue of which each opposes any change in the strength of current flowing through the other by developing an induced e.m.f.Coefficient of mutual inductance (M) of two coils is said to be one henry, when a current change at the rate of 1 ampere/sec. in one coil induces an e.m.f. of one volt in the other coil. The value of M depends on geometry of two coils, distance between two coils, relative placement of two coils etc.The coefficient of mutual inductance of two long co-axial solenoids, each of length l, area of across section A, wound on an air core is … (1)where N1 and N2 are total number of turns of the two solenoids.The mutual inductance M is defined by the equation

N2φ2 = MI1where I1 is the current in coil 1, due to which flux φ2 is linked with each turn of secondary coil.Now we can calculate, e.m.f. e2 induced in secondary by a changing current in first coil. From Faraday‘s lawIf ...(2)The two definitions for M defined by equations (1) and (2) are equivalent. We can express these two equations in words as :- M is numerically equal to the flux-linkage in one circuit, when unit current flows through the other. (we use this definition to calculate M)
- M is numerically equal to the e.m.f. induced in one circuit, when the current changes in the other at the rate of one ampere in each second. (it is used to describe the mutual behavior of two circuits).

For a pair of coils, M12 = M21 = μ0 N1 N2 A/, when wound on one another.

N2φ2 = MI1

- Coefficient of self inductance of two coils in series

- Coefficient of self inductance of two coils in parallel

- The coefficient of coupling between two coils having self inductance L1 & L2 and coefficient of mutual inductance M is
- Generally the value of K is less than 1.
- If K is 1, then the coupling of two coils is tight while if K < 1, then coupling is loose.
- Inductance is pure geometrical factor, and is independent of current or applied e.m.f.
- If the angle between the axis of two closely placed coil is θ then