Question

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.

Answer 1

(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 × $\frac{a}{2}$ = 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
$q=\frac{∆\varphi }{R}$
Here,
ΔΦ = Change in the flux
R = Resistance of the coil
$\therefore q=\frac{∆\varphi }{R}$
$$\begin{gathered} =\frac{B(a^2-0)}{R} \\ =\frac{B{a}^2}{R} \end{gathered}$$