Write Faraday’s laws of electromagnetic induction and obtain an expression of induced e.m.f.

Faraday's laws of electromagnetic induction describe how electric current is generated by changing magnetic fields. Here are the two laws:

### Faraday's First Law of Electromagnetic Induction
**The first law states that a change in the magnetic field within a closed loop induces an electromotive force (e.m.f.) in the loop.**

### Faraday's Second Law of Electromagnetic Induction
**The second law quantifies the induced e.m.f. It states that the magnitude of the induced e.m.f. is proportional to the rate of change of the magnetic flux through the loop.**

Mathematically, the induced e.m.f. (\(\mathcal{E}\)) is given by:
\[ \mathcal{E} = -\frac{d\Phi_B}{dt} \]
where \(\Phi_B\) is the magnetic flux and \(t\) is time. The negative sign in the formula is due to Lenz's Law, which indicates that the direction of the induced e.m.f. is such that it opposes the change in magnetic flux.

### Expression for Induced e.m.f.

To derive the expression for the induced e.m.f., consider a coil of \(N\) turns with a magnetic field \(B\) passing perpendicularly through the coil. Let \(A\) be the area of the coil and \( \Phi_B \) be the magnetic flux through one turn of the coil.

The magnetic flux \(\Phi_B\) through one turn of the coil is given by:
\[ \Phi_B = B \cdot A \]
where \(A\) is the area of the coil and \(B\) is the magnetic field strength.

For \(N\) turns of the coil, the total magnetic flux linkage is:
\[ \text{Total flux linkage} = N \cdot \Phi_B = N \cdot (B \cdot A) \]

The rate of change of this magnetic flux linkage with respect to time is:
\[ \frac{d(\text{Total flux linkage})}{dt} = N \cdot \frac{d(B \cdot A)}{dt} \]

Thus, the induced e.m.f. \(\mathcal{E}\) is:
\[ \mathcal{E} = -N \cdot \frac{d(B \cdot A)}{dt} \]

This expression shows that the induced e.m.f. depends on the number of turns in the coil, the area of the coil, and the rate at which the magnetic field changes.