Cross the field. When does the current stop?
One rod accelerates, cruises, then accelerates again. Follow motion and current across the same field boundaries.
The problem
Smooth rails are 8 m long and 1 m apart, with a 2 Ω resistor joining their left ends. A 0.5 kg rod, 1 m long, starts at rest at x = 0.5 m under a constant 1 N rightward pull. A uniform 1 T field points into the page only between x = 1.5 m and 3.5 m. Measure x from the left rail ends and follow the next 3 s.
Assumptions and boundaries
- The closed loop has negligible rail and rod resistance. Ignore friction, rotation and self-inductance.
- The static field has sharp boundaries along the rails and covers the full rod length. Fringing is neglected.
- The rod moves only rightward and stays on the rails. Use the inside state at entry and the outside state at exit; position and velocity remain continuous.
Questions to explore
- When does current flow? Why does it stop after exit even though flux remains in the loop?
- Must a magnetic field slow the rod? Describe the three stages.
- Where does the energy go? What changes if the exit moves to 2.5 m?
Diagram description
Animated physics diagram: rail_field_crossing; showing time, distance traveled, speed, induced EMF, current, magnetic force (+ right), applied work, kinetic energy, resistive heat.
Unpack one connected process
Determine the active field from position, then use constant acceleration outside and electromagnetic drag inside. Carry position, velocity and accumulated heat across each boundary instead of restarting an independent experiment.
Outside: ma = F; inside: ma = F − B²L²v/R; Wpull = ΔK + Q
1. When does current flow? Why does it stop after exit even though flux remains in the loop?
The rod enters at 1 s and exits at 2 s. Inside, current is 1 A up the rod; before entry and after exit it is zero. After exit the loop still encloses the whole field region, but its flux no longer changes, so there is no induced EMF. Neglecting self-inductance permits a current jump, not a velocity jump.
2. Must a magnetic field slow the rod? Describe the three stages.
Outside, the pull gives acceleration 2 m/s², reaching 2 m/s at 1 s. Inside, magnetic drag is exactly 1 N and balances the pull, so the rod crosses the 2 m field at a constant 2 m/s. After exit at 2 s it accelerates again at 2 m/s², reaching 4 m/s at 3 s. Drag opposes motion, but the net force determines whether speed falls.
3. Where does the energy go? What changes if the exit moves to 2.5 m?
The original rod travels 6 m: applied work is 6 J, final kinetic energy 4 J and Joule heat 2 J. Change only x_max to 2.5 m: exit occurs at 1.5 s, final speed rises to 5 m/s and heat falls to 1 J. Applied work is 7.25 J, equal to kinetic energy 6.25 J plus heat. Each run restarts from the same initial state.
Physics reference: OpenStax · Motional EMF
Test your prediction in code
Move the field exit from 3.5 m to 2.5 m. Compare exit time, final speed and heat.
scene rail_field_crossing
rail track length=8m separation=1m
rod bar mass=0.5kg length=1m x=0.5m
resistor R 2ohm
field B magnetic 1T direction=into_page x_min=1.5m x_max=3.5m
force pull 1N direction=right on bar
bar on track
bar moves through B
track <-> R <-> track
simulate 3s playback=6s
show velocity forces current induction energy heat