State the principle on which a transformer works. Describe the working of a transformer with necessary theory.

A transformer is a device which transforms electric power in one circuit into electric power of same frequency in another circuit.
This works on the principle of mutual inductance between two circuits linked by a common magnetic flux.

Construction & working : A transformer consists of two insulated mutually coupled coils of wire wound on a laminated iron core (or any substance of high magnetic permeability).One of the coils is called ‘primary coil’ of N_P turns and the other one ‘secondary coil’ of N_S turns the primary ‘p’ is connected to a source of emf and the secondary to a load when an alternating e.m.f is applied across the primary coil, the input voltage changes with time. Hence the magnetic flux through the primary also changes with time this changing magnetic flux will be linked with secondary through the core an e.m.f is induced in the secondary
THEORY : Let $\mathrm{\varphi }$ be the flux in each turn in the core at a time t due to current in primary when a voltage v_P is applied to it then by faraday’s law. The emf induced in secondary is
${\mathrm{e}}_{\mathrm{s}}=-{\mathrm{N}}_{\mathrm{S}}\frac{\mathrm{d\varphi }}{\mathrm{dt}}$
The alternating flux $\mathrm{\varphi }$ also induces a back emf in primary given by
${\mathrm{e}}_{\mathrm{p}}=-{\mathrm{N}}_{\mathrm{p}}\frac{\mathrm{d\varphi }}{\mathrm{dt}}$
but ${\mathrm{e}}_{\mathrm{S}}={\mathrm{v}}_{\mathrm{S}}\text{ and }{\mathrm{e}}_{\mathrm{P}}={\mathrm{v}}_{\mathrm{P}}$
$\Rightarrow \frac{{\mathrm{v}}_{\mathrm{s}}}{{\mathrm{v}}_{\mathrm{p}}}=\frac{{\mathrm{N}}_{\mathrm{s}}}{{\mathrm{N}}_{\mathrm{p}}}$
if transformer is 100% efficient that is for then an ideal transformer
output power = input power
$\begin{array}{l}{\mathrm{i}}_{\mathrm{S}}{\mathrm{V}}_{\mathrm{S}}={\mathrm{i}}_{\mathrm{P}}{\mathrm{V}}_{\mathrm{P}}\\ \therefore \frac{{\mathrm{v}}_{\mathrm{s}}}{{\mathrm{v}}_{\mathrm{p}}}=\frac{{\mathrm{N}}_{\mathrm{s}}}{{\mathrm{N}}_{\mathrm{p}}}=\frac{{\mathrm{i}}_{\mathrm{p}}}{{\mathrm{i}}_{\mathrm{s}}}\end{array}$