Flux Linkages and Faraday’s Law of Electromagnetic Induction

Flux Linkages

The product of number of turns (N) of the coil and magnetic flux \((\phi \)linking the coil is called flux linkages.

$$ \text {i.e. Flux Linkage} = N \phi $$

Experiments show that the magnitude of e.m.f induced in a coil is directly proportional to the rate of change of flux linkages. If N be the number of turns of the coil and the magnetic flux linking the coil changes from \(\phi _1\) to \(\phi _2\) in time t, then

$$\begin{align*}\text {induced e.m.f} \: (\epsilon) \: \alpha \: \text {Rate of change of flux linkages} \\ \therefore \: \epsilon \propto \frac {N\phi _2 – N\phi _1}{t} \\ \end{align*}$$

Magnetic Flux Density (β)

The number of magnetic lines of force passing through unit area normally or magnetic flux per unit area is called magnetic flux density.

$$\begin{align*} B &= \frac {\phi }{A_1} \\ \phi &= BA_1 \\ &= BA\cos \theta \\ \therefore \phi &= \vec B .\vec A \\ \end{align*}$$

Special cases

1. If \(\theta = 0 \) i.e. flux is normal to the surface then \( \phi _{\text {max}} = BA \).
2. If \(\theta = 90 \) i.e. flux is parallel to the surface then \( \phi _{\text {min}} = 0 \).

Faraday’s Law of Electromagnetic Induction

There were various series of an experiment performed by Faraday to prove electromagnetic induction. Faraday’s laws are as follows:

1. Whenever there is a change in the magnetic flux associated with a close lobe, there is an induced e.m.f or current and the e.m.f remains as long as the change in flux takes place.
2. The rate of change of magnetic flux is directly proportional to the rate of change of flux.
   $$\begin{align*} \text {i.e} \: \:\text {Induced e.m.f} \: \epsilon \propto \text {rate of change of flux linkage.} \\ \epsilon &\propto \frac {N\phi _2 – N\phi _1}{t} \\ \text {or,} \: \epsilon &= k\frac {(N\phi _2 – N\phi _1)}{t} \\ \text {where k is proportionality constant.} \\ \text {In SI-system,} \: K = 1 \\ \epsilon &= \frac {N\phi _2 – N\phi _1}{t} \\ \text {In differential form, we have,} \\ \epsilon &= N \frac {d\phi }{dt} \dots (i)\\ \end{align*}$$

Since direction of induced e.m.f. is acting in the opposite direction to the cause due to which it is produced. So equation (i) can be written as

$$ \epsilon =- N \frac {d\phi }{dt} $$

which is combined Faraday-Lenz law or simply Faraday’s law.

Lenz’s Law

Lenz’s law states that the induced e.m.f opposes the motion or change producing it. In other words, Lenz’s law states that the induced current opposes the flux change causing it.

Consider a case in which a current is induced in a coil due to change in magnetic flux. For this assume that N-pole of a magnet is approaching towards the end of the coil. If len’s law applies, the induced current should flow in the coil that makes near end of coil behave as N-pole and repels the N-pole of a magnet. So, opposes the flux change. In the same arrangement, if the magnet is moved away from the coil, the near end of the coil should behave like a S-pole and attract the N-pole of the magnet. So, opposes the flux change again. The induced current in the case is therefore in the opposite direction to that when the magnet approaches. Lenz’s law is added in the expression of Faraday’s law by inducing a negative sign which shows that the current due to induced e.m.f opposes flux change.

Lenz law is in accordance with the law of conservation of energy

If S-pole were produced at the near end of the coil when N-pole of the magnet is approaching then the magnet would accelerate towards the coil, gaining kinetic energy as well as generating electrical energy. Thus, energy will be created. However, work has to be done to overcome the forces. Thus mechanical energy is transferred into electrical energy.