Conducting Rod on Parallel Rails Connected to a Capacitor

Problem Statement

Two sufficiently long, smooth parallel metal rails are fixed horizontally at an interval of $L$L. An uncharged capacitor with capacitance $C$C is connected at one end of the rails. A conducting rod of mass $m$m and electrical resistance $R$R is placed perpendicular to the rails. The electrical resistance of the rails is negligible. A uniform magnetic field of magnetic flux density $B$B acts vertically downward throughout the space.

At time $t$t, the rod is given an initial velocity $v_0$v0​ parallel to the rails. The rod moves while always staying perpendicular to the rails, and after sufficient time, it reaches a constant speed $v_f$vf​.

Find the final speed $v_f$vf​ of the rod.

(Coordinate system: the $x$x-axis is along the rails, with the direction of $v_0$v0​ as positive. The $z$z-axis is positive upward. The counterclockwise current (when seen from above) is defined as positive.)

Constraints

• $L = 0.50 \text{ m}$L=0.50 m
• $C = 2.0 \times 10^{-3} \text{ F}$C=2.0×10−3 F
• $m = 1.2 \times 10^{-2} \text{ kg}$m=1.2×10−2 kg
• $R = 0.10 \text{ } \Omega$R=0.10 Ω
• $B = 2.0 \text{ T}$B=2.0 T
• $v_0 = 10 \text{ m/s}$v0​=10 m/s

Input Format

The final speed $v_f$vf​ is expressed as an irreducible fraction $\frac{p}{q} \text{ (m/s)}$qp​ (m/s). Find the value of $p + q$p+q.