A wire of mass m and length I can freely slide on a pair of parallel, smooth, horizontal rails placed in a vertical magnetic field B . The rails are connected by a capacitor of capacitance C. The electric resistance of the rails and the wire is zero. If a constant force F acts on the wire as shown in the figure, find the acceleration of the wire.

Options

A. `a = (C^(2) B^(2) l - F)/(m)`
B. `a = (F)/(m + CBl)`
C. `a = (F C^(2) B^(2) l )/(m)`
D. `a = (F)/(m + c B^(2)l^(2))`

Answer

D

Explanation

Let at time t, `upsilon` be the velocity of wire.
EMF induced, `epsilon = B l upsilon`
Charge (in magnitude) appearing on the plates of capacitor is
`q = C epsilon = CB l upsilon`
Now current, `I = (dq)/(dt) = CB l (dv)/(dt) = CB la`
[ `:. (d upsilon)/(st)` = a. i.e., acceleration]
Force on wire due to the current is
`F' = IlB = (CB la) lB = CB^(2) l^(2) a`
The net force in the direction of motion of conductor is `F - F' = ma` or `F = ma + CB^(2) l^(2) a`
or `a = (F)/(m + CB^(2) l^(2))`