A square metal wire loop of side $$10\,cm$$  and resistance 1 ohm is moved with a constant velocity $${v_0}$$ in a uniform magnetic field of induction $$B = 2\,{\text{weber}}/{m^2}$$    as shown in the figure. The magnetic field lines are perpendicular to the plane of the loop (directed into the paper). The loop is connected to a network of resistors each of value 3 ohms. The resistances of the lead wires $$OS$$ and $$PQ$$ are negligible. What should be the speed of the loop so as to have a steady current of 1 milliampere in the loop ? Give the direction of current in the loop.

Solution :

The network behaves like a balanced wheatstone bridge.
The induced emf developed is given by
$$\eqalign{ & e = vB\ell = v \times 2 \times 0.1 = 0.2v\,......\left( {\text{i}} \right) \cr & {\text{Now,}}\,\,e = IR \cr & e = {10^{ - 3}} \times 4 = 4 \times {10^3}{\text{amp}}\,......\left( {{\text{ii}}} \right) \cr} $$
From (i) and (ii),
$$\eqalign{ & 0.2\,v = 4 \times {10^{ - 3}} \cr & \therefore v = \frac{{4 \times {{10}^{ - 3}}}}{{0.2}} = 0.02\,m/s \cr} $$