Question

Derive Faradays law of induction from law of conservation of energy.

Answer 1

Consider a situation shown in figure a rod of length $\left(l\right)$ slide over two rail S R rod P Q is moving with speed V towards right. The electromagnetic force act on the rod is $f_m=\frac{B^2{l}^{2}V}{R}$ towards R is the resistance of the circuit then the work done by the mechanical movement is $W_m=f_m.dx$

Mechanical power develop $F_m=\frac{B^2{l}^{2}V}{R}$

$p_{mech}=-f_m\frac{dx}{dt}={-f}_{m}V=\frac{{-B}^2{l}^2{V}^2}{R}\to \left(1\right)$

If current induced in the circuit due to induced emf. e then electrical power developed is $p_e=\frac{{\epsilon }^2}{R}\to \left(ii\right)$

If energy is conserve then $p_e=p_{mech}$ using equation (1)

$-\frac{B^2{l}^2{V}^2}{R}=\frac{{\epsilon }^2}{R}\epsilon =-BlV\to \left(iii\right)$

We know that flux $\varphi =\int B.ds=B.lx$

Area of PQRS = $lx$

Now equation (iii) written as $\epsilon =-Bl\frac{dx}{dt}=\frac{-d}{dt}Blx=\frac{-d\varphi }{dt}$

Here we can see that induced emf

$\epsilon =\frac{-d\varphi }{dt}$ that is Farady's law.