A circular conducting loop of radius r_oand having resistance per unit length $λ$ as shown in the figure is placed in a magnetic field B which is constant in space and time. The ends of the loop are crossed and pulled in opposite directions with a velocity v such that the loop always remains circular and the radius of the loop goes on decreasing, then which of the following is not true

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Radius of the loop changes with r as $r = r_{0} - vt / π$

Current induced in the loop is $I = \frac{Bv}{2 πλ}$

Current induced in the loop is $I = \frac{Bv}{πλ}$

EMF induced in the loop as a function of time is $e = 2 Bv [r_{0} - vt / π]$

answer is C.

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Detailed Solution

Perimeter is decreasing at a rate of 2v.
$\begin{array}{l} ∴ \frac{d}{dt} (2 πr) = 2 v \Rightarrow \frac{dr}{dt} = \frac{v}{π} ∴ r = (r_{0} - \frac{v}{π} t) \\ ϕ = B \cdot {πr}^{2} \\ \Rightarrow ε = |\frac{- dϕ}{dt}| = B \cdot 2 π \cdot r \frac{dr}{dt} \\ ε = 2 Bπ (r_{0} - \frac{v}{π} t) \cdot \frac{v}{π} = 2 Bv (r_{0} - \frac{v}{π} t) \\ Thus, I = \frac{ε}{R} = \frac{2 Bvr}{λ \cdot 2 πr} = \frac{Bv}{πλ} \end{array}$

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