Electromagnetic Induction

The phenomenon of production of an electromotive force (emf) in a coil due to a change in magnetic flux linked with the coil is called Electromagnetic Induction. The emf produced in the coil is called the induced emf, and the current that flows in the coil due to the induced emf is called the induced current.

Magnetic Flux

The magnetic flux through any surface area is defined as the total number of magnetic lines of force passing normally through the surface. It is denoted by ϕ. It is a scalar quantity. Its SI unit is weber (Wb) or tesla metre² (Tm²), and the CGS unit is Maxwell.

1 Maxwell=10⁻⁸ weber

Consider a small surface area A placed in a uniform magnetic fieldB. Let θ be the angle between the normal to the surface and the magnetic field B as shown in the figure.

The magnetic flux through the surface is measured by the dot product of the magnetic field B and the area vector A.

Consider a small surface area A placed in a uniform magnetic field B. Let θ be the angle between the normal to the surface and the magnetic field B (as shown in the figure). The magnetic flux through the surface is measured by the dot product of the magnetic field and the area vector A. Thus, the magnetic flux through the surface is:

Special cases:

I. When θ=0∘, i.e., if the surface lies perpendicular to the direction of the magnetic field:

ϕ=BAcos⁡0=BA(maximum)
Thus, magnetic flux through a given surface is maximum when θ=0∘

II. When θ=90∘, i.e., if the surface lies parallel to the direction of the magnetic field:
ϕ=BAcos⁡90∘=0(minimum)

Thus, magnetic flux through a given surface is zero when θ=90∘

Dimensional formula for magnetic flux:

We have, Magnetic flux ϕ=BAcos⁡θ.  Since B=(F/qV)​ and cos⁡θ is dimensionless, so.

Magnetic Flux Density:
It is defined as magnetic flux passing per unit area normal to the direction of flux. Magnetic flux density is given by B=(ϕ/A). Its SI unit is Wb m−2or Tesla (T), and its CGS unit is Gauss (G).

1T=10⁴G

Flux Linkage:

It is the total value of flux through the whole area of a coil. If a coil placed in a magnetic field has $N$ turns, each having an area $A$, then the flux linkage with the coil is given by ϕ′=Nϕ where $\varphi$ is the flux linked with each turn. Thus, Flux linkage =NABcos⁡θ

Faraday’s Laws of Electromagnetic Induction:

Based on a series of experiments, Michael Faraday gave the following laws of electromagnetic induction:

1. First Law: Whenever the magnetic flux linked with a coil changes, an emf is induced in it.
2. Second Law: The induced emf lasts as long as the change in magnetic flux continues.
3. Third Law: The magnitude of the induced emf in the coil is directly proportional to the rate of change of magnetic flux linked with the coil.

If the magnetic flux linked with a coil changes from ϕ1​ to ϕ2​ in time $t$, and the induced emf is $E$, then according to Faraday’s law:

where $K$ is a proportionality constant whose value in SI units is 1. The negative sign indicates that the direction of the induced emf is such as to oppose the change in flux (Lenz’s Law).

For $N$ the number of turns of the coil, the induced emf is

Direction of Induced Emf:

The direction of induced emf (or induced current) can be determined by:

1. Lenz’s Law
2. Fleming’s Right-Hand Rule

Lenz’s Law

It states that “the direction of induced current is such that it opposes the cause producing it.”

Consider a bar magnet and a coil.

• When the N-pole of the bar magnet is moved toward the coil, the magnetic flux linked with the coil changes, and an emf is induced, causing current to flow in an anticlockwise direction. This produces a magnetic field in the coil with its N-pole toward the magnet, so the approaching magnet is repelled.

• When the N-pole of the bar magnet is moved away from the coil, the current flows in a clockwise direction, producing a magnetic field with an S-pole toward the left, which attracts the receding magnet.

Lenz’s Law in Accordance with the Principle of Conservation of Energy

When the magnet is brought near or taken away from the coil, the flux linked with the coil changes, causing an emf to be induced. The direction of the induced emf is such that it opposes the motion of the approaching or receding magnet.
In this process, mechanical work is done to move the magnet, which is converted into electrical energy in the coil. The induced current produces a magnetic field that causes the magnet to be repelled or attracted, requiring further mechanical work. Thus, mechanical energy is converted into electrical energy.

This aligns with the principle of conservation of energy, which states that energy cannot be created or destroyed but only transformed from one form to another.

Fleming’s Right-Hand Rule

Fleming’s Right-Hand Rule is used to determine the direction of induced current. It states:

“If the thumb, forefinger, and middle finger of the right hand are stretched mutually perpendicular to each other such that the forefinger points in the direction of the magnetic field, the thumb points in the direction of motion of the conductor, then the middle finger points in the direction of the induced current.”

Numericals

1. A flat coil of radius 8 mm has 500 loops of wire in it. It is placed in a magnetic field of 0.30 T so that maximum flux goes through it. It is rotated to a position such that no flux goes through it in 0.02 sec. Find the average induced emf between the terminals of the coil.

2. The magnetic flux passing perpendicular to the plane of a coil is given by ϕ=4t2+5t+2ϕ, where ϕis in Weber and  t is in seconds. Calculate the instantaneous emf induced in the coil when t=3 sec.

The magnitude is 29 V, and the negative sign indicates the direction (opposing the change in flux).

Method of Producing Induced emf: