Conducting rods moving in a magnetic field and a capacitor circuit

Problem Statement

Two sufficiently long, smooth conductive rails are placed parallel to each other in a horizontal plane, separated by a distance $L$L. A capacitor with capacitance $C$C and a resistor with resistance $R$R are connected in series at the left end of each rail. A uniform magnetic field (magnetic flux density $B$B) is applied vertically upward, perpendicular to the rail surface, in the space containing the rails.

Now, a conducting rod of mass $m$m is placed perpendicular to the rails. Starting from a stationary position at time $t=0$t=0, a constant external force of magnitude $F$F is applied parallel to the rails and directed to the right, causing the conducting rod to be pulled. The resistance of the conducting rod and rails, the self-inductance of the circuit, and friction and air resistance are all negligible. The capacitor has no initial charge.

After a sufficient amount of time has elapsed since the conducting rod was started to be pulled, the magnitude of the current flowing through the circuit approaches a constant value $I$I. Find the algebraic expression for this magnitude $I$I.

Constraints

• $F = 3.00 \text{ N}$F=3.00 N
• $m = 0.200 \text{ kg}$m=0.200 kg
• $C = 0.500 \text{ F}$C=0.500 F
• $B = 2.00 \text{ T}$B=2.00 T
• $L = 0.400 \text{ m}$L=0.400 m
• $R = 10.0 \text{ }\Omega$R=10.0 Ω

Input Format

The magnitude of the current $I \text{ [A]}$I [A] after a sufficient amount of time has elapsed is expressed as an irreducible fraction $\frac{p}{q}$qp​ (where $p and q$pandq are relatively prime natural numbers). Enter the value of the sum of the numerator and denominator, $p + q$p+q.