A conducting wire of mass m is pulled with a constant force F with uniform velocity $v_{0}$ on two parallel rails lying in horizontal plane and separated by distance $l$ . Ends of rails are connected by a coil of self inductance L. At time t=0, $x = x_{0} (≃ 0)$ and a uniform vertical magnetic field B is present in space. When the conducting wire is at distance $x$ from the initial position,

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Force F is directly proportional to $x^{2}$

Energy stored in coil is directly proportional to $x^{- 2}$

Energy stored in coil is directly proportional to x

Force F is directly proportional to x

answer is A.

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Detailed Solution

$\begin{array}{l} L \frac{d l}{d t} = B l \frac{d x}{d t} \\ \Rightarrow \int_{0}^{l} d l = \frac{B l}{L} \int_{0}^{l} d x \\ I = \frac{B l}{L} x \\ U = \frac{1}{2} L I^{2} \Rightarrow U \propto x^{2} \\ F = I B l \\ F = \frac{B^{2} l^{2}}{L} x \\ F \propto x \end{array}$