Difficult inductance + torque rolling rail problem

[oreosama]

Homework Statement

Homework Equations

$$F = I\vec{l} \times \vec{B}$$
$$\varepsilon = BLv$$

The Attempt at a Solution

Every direction i go seems real wrong


$$F = I\vec{l} \times \vec{B}$$

$$F = \frac{\varepsilon \vec{l} \times \vec{B}}{R}$$
$$F = \frac{\varepsilon LB}{R}$$
$$ma = \frac{VLB}{R}$$
$$v = \frac{VLB}{mR}t$$


while the math seems to all be good here, this implies like every answer after is going to be infinity which seems plain wrong.. I assume I should havea current from the battery versus a current induced and thus have two forces. but things get messy and I can never get a v just in terms of t when I mess with it in this direction. thanks for any help

 

[haruspex]

I assume I should havea current from the battery versus a current induced and thus have two forces.

Not quite. The motion of the cylinder through the field will induce a "back emf". I'm no expert in this area, but my guess is that the cylinder will cease to accelerate when the back emf equals and cancels the applied emf.