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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.
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
\[\therefore q = \frac{∆ \phi}{R}\]
\[= \frac{B( a^2 - 0)}{R}\]
\[ = \frac{B a^2}{R}\]
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