Conducting Rod Moving in a Magnetic Field Connected to a Capacitor

Problem Statement

On a horizontal plane with a uniform magnetic field of magnetic flux density $B$B directed vertically upward, two sufficiently long parallel metal rails are fixed at an interval of $L$L. A capacitor of capacitance $C$C (initially uncharged) is connected at the left end of the rails.

A metal rod of mass $m$m is placed perpendicular to the rails, and at time $t=0$t=0 it is given an initial velocity $v_0$v0​ to the right. The metal rod moves smoothly while remaining perpendicular to the rails. The electrical resistance of the entire circuit and the friction between the rod and rails are all negligible.

After sufficient time has passed, the velocity of the metal rod asymptotically approaches a constant value $v_f$vf​. Find this final velocity $v_f$vf​.

Constraints

• Mass of metal rod $m = 2.0 \times 10^{-2}\text{ kg}$m=2.0×10−2 kg
• Capacitance of capacitor $C = 0.50\text{ F}$C=0.50 F
• Magnetic flux density $B = 0.20\text{ T}$B=0.20 T
• Rail spacing $L = 1.0\text{ m}$L=1.0 m
• Initial velocity $v_0 = 12\text{ m/s}$v0​=12 m/s

Input Format

Find the final speed $v_f$vf​ of the metal rod in $\text{m/s}$m/s and give the answer as a positive integer.