Revision: 12th Std >> Electromagnetic Induction MAH-MHT CET (PCM/PCB) Electromagnetic Induction

The phenomenon in which electric current is generated in a conductor or closed coil due to a varying magnetic field is called electromagnetic induction.

If the current-carrying conductor is held in the right hand such that the thumb points in the direction of the current, then the direction of the curl of the fingers will give the direction of the magnetic field.

Whenever the number of magnetic lines of force (magnetic flux) passing through a coil changes, an electric current is induced in the coil. This current is called the induced current.

Magnetic flux (Φ_B​) is the total number of magnetic field lines passing normally through a given area.

Flux of a vector field through a surface is defined as the total number of field lines passing normally through that surface. Mathematically, it is the scalar (dot) product of the field vector and the area vector.

Electric flux (Φ_E​) through a surface is the measure of the electric field lines passing through that surface.

The emf induced across the ends of a conductor due to its motion in a magnetic field is called motional emf.

Eddy currents are circulating currents induced in the bulk of a conductor when the magnetic flux linked with it changes.

They are named after the whirlpool-like eddies formed in flowing water.

It is defined as magnetic flux linked with the solenoid when unit current flows through it.

The self-inductance of a circuit is the ratio of magnetic flux (produced due to current in the circuit) linked with the circuit to the current flowing in it. 

The property of a coil by which it opposes the change in its own current and induces an emf in itself — numerically equal to the ratio of magnetic flux (produced due to current in the circuit) linked with the circuit to the current flowing in it, or the ratio of induced emf produced around the circuit to the rate of change of current in it — is called self-inductance.

OR

The self-inductance (L) of a coil is defined as the ratio of the total magnetic flux linkage through the coil to the current flowing in it. Equivalently, it equals the magnitude of the induced EMF per unit rate of change of current.

The potential energy accumulated in an inductor or magnetic system due to the presence of the magnetic field is called the energy stored in a magnetic field.

The energy stored per unit volume in the magnetic field is called energy density.

The mutual inductance (M) of two circuits (or coils) is the magnetic flux (Φ_s) linked with the secondary circuit per unit current (I_P) of the primary circuit.

The coefficient of coupling K between two coils is the fraction of the total magnetic flux produced by one coil that links with the other coil.

The property of two coils by which a change in current in one coil induces an emf in the other coil — equal to the magnetic flux linked with one circuit per unit current in the other, or the value of induced emf produced in the secondary circuit per unit rate of change in current in the primary circuit — is called mutual inductance.

OR

Mutual Inductance (M) of a pair of coils is defined as the ratio of the total magnetic flux linkage in the secondary coil to the current in the primary coil that produces it.

The transformer is a device used for converting low voltage into high voltage and high voltage into low voltage. It works on the principle of electromagnetic induction.

An electrical device which converts low alternating voltage at high current to high alternating voltage at low current (or vice versa) — i.e., a device which reduces or increases the voltage in an AC circuit through mutual induction — is called a transformer.

For a uniform field \[\vec F\] passing through a flat surface of area A, inclined at angle θ to the field:

For a non-uniform field or curved surface, flux is expressed as a surface integral:

\[\Phi_E=\int_A\vec{E}\cdot d\vec{A}\]

For a closed surface (used in Gauss's Law):

\[\Phi_E=\oint\vec{E}\cdot d\vec{A}=\frac{q_{enc}}{\varepsilon_0}\]

e = Blv

• B = magnetic field
• l = length of conductor
• v = velocity

L = \[\frac{N\Phi_B}{I}\]

ε = -L\[\frac{dI}{dt}\]

Where:

N₂​ϕ₂₁ ​∝ I_1​ ⟹ N_2​ϕ_21​ = M ⋅ I₁

Therefore:

M = K\[\sqrt {L_1L_2}\]

Where:

• L₁, L₂​ = Self-inductances of coil 1 and coil 2
• K = Coefficient of coupling (dimensionless, no units)
• Range: 0 ≤ K ≤ 1

Therefore:

M ≤ \[\sqrt {L_1​L_2}\]​​

\[\frac{V_s}{V_p}=\frac{N_s}{N_p}=k\]

Where:

• V_s = Secondary (output) voltage (V)
• V_p = Primary (input) voltage (V)
• N_s = Number of turns in secondary coil
• N_p = Number of turns in primary coil
• k = Transformation ratio (turns ratio)

\[\frac{I_p}{I_s}=\frac{N_s}{N_p}=\frac{V_s}{V_p}\]

\[\frac{V_s}{V_p}=\frac{N_s}{N_p}=\frac{I_p}{I_s}=k\]

 P_input = P_output

First law: Whenever there is a change of magnetic flux in a closed circuit, an induced emf is produced in the circuit. This law is a qualitative law as it only indicates the characteristics of induced emf.

Second law: The magnitude of the induced emf produced in the circuit is directly proportional to the rate of change of the magnetic flux linked with the circuit. This law is known as the quantitative law, as it gives the magnitude of the induced emf.

Faraday’s First Law: Whenever the magnetic flux linked with a circuit changes, an emf is induced in the circuit.

Faraday’s Second Law: The magnitude of the induced emf is equal to the rate of change of magnetic flux.

e = `-(d phi)/dt`

For a coil of N turns:

e = `-N (d phi)/dt`

Negative sign indicates Lenz’s law (direction opposes cause).

It is stated that the direction of induced e.m.f. is always in such a direction that it opposes the change in magnetic flux.

e = `(d phi)/(dt)`

Consider a rectangular metal coil PQRS. Let ‘L’ be the length of the coil. It is placed in a partly magnetic field ‘B’. The direction of the magnetic field is perpendicular to the paper and into the paper. The ‘x’ part of the coil is in the magnetic field at instant t. If the coil is moved towards the right with a velocity v = `dx/dt` with the help of an external agent, such as a hand. The magnetic flux through the coil is:

Φ = BA = BLx

∴ Φ = BLx     ...(1)

There is relative motion of a current through the coil. Let ‘i’ be current through the coil.

Three forces act on the coil.

F₁ on conductor PL ∴ F₁ = B_i x, vertically upward.

F₂ on conductor MS ∴ F₂ = B_i x, vertically downward.

F₃ on conductor SP ∴ F₃ = B_i L towards left.

F₁ and F₂ are equal and opposite and also on the same line. They will cancel each other; F₃ is a resultant force. The external agent has to do work against this force.

∴ F₃ = −B_i l    ...(−ve sign indicates that force is opposite to dx.)

If dx is the displacement in time dt, then the work done (d_w) = F₃ dx.

∴ d_w = − B_iL dx

This power is an electrical energy ‘ei’ where ‘e’ is an induced e.m.f.

∴ e_i = `-(B_i ldx)/(dt)`

∴ e = `-(BLdx)/(dt)`

∴ e = −BLv

∴ e = `-d/dt (BLx)`

∴ e = `(-d phi)/(dt)`    ...[from eq (1)]

Lenz’s Law states that the direction of the induced electromotive force (EMF) and the resulting current in a conductor is always such that it opposes the change in magnetic flux that caused it. 

Mathematically, Lenz’s Law is expressed as:

ε = `(-d phi_B)/dt`

Where,

ε = Induced EMF

Φ_B = Magnetic flux

The negative sign indicates opposition to the change in flux.

Statement:

The direction of the induced emf, or the induced current, in any circuit is such as to oppose the cause that produces it. This law is known as Lenz’s Law.

Explanation / Proof:

• When the north pole of a magnet is moved towards the coil, an induced current flows in the coil in such a direction that the near (left) face of the coil behaves like a north pole.
• Due to the repulsion between the like poles, the motion of the magnet towards the coil is opposed.
• When the north pole of the magnet is moved away from the coil, the induced current flows in such a direction that the near face of the coil becomes a south pole.
• The attraction between opposite poles then opposes the motion of the magnet away from the coil.

In both cases, the induced current opposes the magnet's motion, which is the cause of the current. Therefore, work has to be done to move the magnet, and this mechanical work appears as electrical energy in the coil.

Direction of Induced Current (Fleming’s Right-Hand Rule):

• Stretch the right-hand thumb, forefinger, and middle finger so that they are mutually perpendicular.
• The forefinger points in the direction of the magnetic field.
• The thumb points in the direction of motion of the conductor.
• The middle finger then gives the direction of the induced current.

Conclusion:

Lenz’s Law shows that the induced current always acts in such a direction as to oppose the cause that produces it. This ensures that mechanical energy is converted into electrical energy, and no energy is produced without work being done.

Statement:

When the magnetic flux through a circuit is changing, an induced electromotive force (emf) is set up in the circuit whose magnitude is equal to the negative rate of change of magnetic flux. This is also known as Neumann’s Law.

Mathematical Expression:

If ΔΦ_B​ is the change in magnetic flux in a time interval Δt, then the induced emf e is given by:

e = \[-\frac{\Delta\Phi_B}{\Delta t}\]​​

In the limiting case as Δt → 0:

e = \[-\frac{d\Phi_{B}}{dt}\]​​

• If dΦ_B is in weber (Wb) and dtdtdt in seconds (s), then the emf eee will be in volts (V).
• This equation represents an independent experimental law, which cannot be derived from other experimental laws.

For a tightly-wound coil of N turns, the induced emf becomes:

e = \[-N\frac{d\Phi_B}{dt}\] or e = \[-\frac{d(N\Phi_B)}{dt}\]​

Here, NΦ_B is called the ‘number of magnetic flux linkages’ in the coil, and its unit is weber-turns.

Explanation:

Consider a magnet and a coil:

• When the north pole of a magnet is near a coil, a certain number of magnetic flux lines pass through the coil.
• If either the coil or the magnet is moved, the number of magnetic flux lines (i.e., the magnetic flux) through the coil changes.

Cases:

• Magnet moved away from the coil → Decrease in magnetic flux through the coil.
• Magnet brought closer to the coil → Increase in magnetic flux through the coil.

In both cases, an emf is induced in the coil during the motion of the magnet.

• Faster motion → Greater rate of change of flux → Higher induced emf.
• If both the magnet and coil are stationary, or both are moving in the same direction with the same velocity, there is no change in flux → No induced emf.

Special Case:

• If the coil is an open circuit (i.e., infinite resistance), emf is still induced, but no current flows.
• This shows that it is the change in magnetic flux that induces emf, not current.

Conclusion:

Neumann’s Law establishes that a changing magnetic flux through a circuit induces an emf, and the induced emf is proportional to the rate of change of flux, with a negative sign indicating the direction (as per Lenz’s law).

The direction of the induced current in a circuit is such that the magnetic field produced by the induced current opposes the change in the magnetic flux that induces the current; equivalently, "the direction of the induced emf or induced current in a circuit is such that it opposes the change in magnetic flux which produces it." The direction of induced emf is the same as that of induced current. This law is based on the law of conservation of energy.
Illustration with a bar magnet:

• When a magnet approaches a coil, an anti-clockwise current is induced in the coil.
• When a magnet is taken away from a coil, a clockwise current is induced in the coil.

Let a current I_P flow through the circular loop of radius R. The magnetic induction at the centre of the loop is

B_P = `(mu_0I_P)/(2R)`

As, r << R, the magnetic induction B_P may be considered to be constant over the entire cross-sectional area of the inner loop of radius r. Hence magnetic flux linked with the smaller loop will be

`Φ_S = B_PA_S = (mu_0I_P)/(2R)pir^2`

Also, Φ_S = MI_P

∴ M = `Phi_S/I_P = (mu_0pir^2)/(2R)`

Statement: The mutual inductance of coil 1 with respect to coil 2 equals the mutual inductance of coil 2 with respect to coil 1.

This is called the Reciprocity Theorem of Mutual Inductance.

Implication: It does not matter which coil drives the current — the mutual inductance M between the pair is always the same property of the system, not just one coil.youtube

A transformer is based on the principle of mutual induction, i.e., whenever the magnetic flux linked with a coil changes, an emf is induced in the neighbouring coil. For an ideal transformer there is no loss of power, so P_input = P_output​. On the basis of winding, transformers are of two types — step-up and step-down.

• Electricity and magnetism are interrelated.
• Electric current can produce a magnetic field.
• A changing magnetic field can produce electric current.
• The production of electric current by a varying magnetic field is called electromagnetic induction.
• Michael Faraday showed in 1831 that a moving magnet can produce current in a conductor.
• The current produced due to electromagnetic induction is called induced current.
• Generators, transformers, induction motors, and wireless chargers work on electromagnetic induction.