Figure shows a metallic square frame of edge a in a vertical plane. A uniform magnetic field B exists in the space in a direction perpendicular to the plane of the figure. Two boys pull the opposite corners of the square to deform it into a rhombus. They start pulling the corners at t = 0 and displace the corners at a uniform speed u. (a) Find the induced emf in the frame at the instant when the angles at these corners reduce to 60°. (b) Find the induced current in the frame at this instant if the total resistance of the frame is R. (c) Find the total charge which flows through a side of the frame by the time the square is deformed into a straight line.

#### Solution

(a) The effective length of each side is the length that is perpendicular to the velocity of the corners.

Thus, the effective length of each side is a sin θ.

Net effective length for four sides = 4 × `a/2` = 2a

∴ Induced emf = Bvl = 2Bau

(b) Current in the frame is given by

\[i= \frac{e}{R} = \frac{2auB}{R}\]

(c) Total charge q, which flows through the sides of the frame, is given by

\[q = \frac{∆ \phi}{R}\]

Here,

ΔΦ = Change in the flux

R = Resistance of the coil

\[\therefore q = \frac{∆ \phi}{R}\]

\[= \frac{B( a^2 - 0)}{R}\]

\[ = \frac{B a^2}{R}\]