What is induced emf? Write Faraday’s law of electromagnetic induction. Express it mathematically. A conducting rod of length L, with one end pivoted, is rotated with a uniform angular speed ‘ω’ in a vertical plane, normal to a uniform magnetic field ‘B’. Deduce an expression for the emf induced in this rod. 

Whenever the magnetic flux linked with a closed circuit changes, an emf is set up across it which lasts only as long as the change in flux is taking place. This emf is called induced emf. 

According to Faraday's law of electromagnetic induction, the magnitude of induced emf is equal to the rate of change of magnetic flux linked with the closed circuit (or coil).

Mathematically,
                      $\mathrm{E} =-\mathrm{N}\frac{{\mathrm{d\varphi }}_{\mathrm{B}}}{\mathrm{dt}}$ 

where,
N is the number of turns in the circuit and,
Φ_B is the magnetic flux linked with each turn. 

Suppose the conducting rod completes one revolution in time T.
Then,
Change in flux = B x Area swept
                     = B x $\mathrm{\pi }$l²
Therefore, 

              <sup>$\mathrm{Induced} \mathrm{emf} = \frac{\mathrm{Change} \mathrm{in} \mathrm{flux}}{\mathrm{Time}}$ 

                                  $\mathrm{\epsilon } = \frac{\mathrm{B}\times {\mathrm{\pi l}}^2}{\mathrm{T}}$</sup>

But,                     $\mathrm{T} = \frac{2\mathrm{\pi }}{\mathrm{\omega }}$ 

$\therefore$                     $$\begin{gathered} \mathrm{\epsilon } = \frac{\mathrm{B}\times \mathrm{\pi }{l}^2}{2\mathrm{\pi }/\mathrm{\omega }} \\ = \frac{1}{2}{\mathrm{Bl}}^2\mathrm{\omega }. \end{gathered}$$