A wire of mass $m$ and length $l$ is placed on a smooth incline making an angle $\theta$ with the horizontal, whose front view is shown in figure. When a finite amount of charge is passed through it in an infinitesimal time, the wire immediately acquires some velocity and then ascends the incline by a distance $s$. For this small duration, we can neglect the gravitational force because the current can be considered very large due to small time duration. The amount of charge passed through the wire is 

When a charge is passed through a wire, a current flows in the wire (for a very small time ). As a result of this, the wire experience a magnetic force for a very small duration, i. e during the time for which charge has been passed through the wire, due to which it acquires some velocity, then onwards, it moves under gravity. For the wire to move up incline, the positive charge has to pass through the wire in a direction coinciding with into plane of paper. 

$$\begin{gathered} F_m=IBl=\frac{dq}{dt}\times Bl=ma \\ \Rightarrow m\frac{dv}{dt}=\frac{dq}{dt}\times Bl\Rightarrow mdv=dq\times Bl \\ \Rightarrow mv=q\times Bl \\ \Rightarrow v=\frac{q\times Bl}{m}\Rightarrow q=\frac{mv}{Bl} \end{gathered}$$

Where q is the charge passed through the wire and v is the velocity acquired by the wire just after charge had been passed through it. 
From kinematics  $v^2=2g\sin \theta s$
$q=\frac{m\sqrt{2gs\sin \theta }}{Bl}$