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Upper secondary · Newton's laws · Induction · Energy

The rod that gets harder to accelerate

One rod links motion, current and heating. Why doesn't a constant pull produce constant acceleration?

The problem

Two smooth horizontal rails are 6 m long and 1 m apart, joined at the left by a 2 Ω resistor. A 0.5 kg conducting rod, 1 m long, bridges the rails at 0.5 m from their left ends. A uniform 1 T magnetic field points into the page. Starting from rest, pull the rod rightward with a constant 1 N force for 3 s.

Assumptions and boundaries
  • The rod and rails have negligible resistance; only the fixed resistor dissipates heat. The loop stays closed.
  • Ignore friction, self-inductance and air resistance. The uniform field covers the whole motion region.
  • The rod stays perpendicular to the rails. Leaving the rails or crossing a field boundary is not modeled.

Questions to explore

  1. Which way do the induced current and magnetic force point?
  2. Why does acceleration fade? Find speed and current at 3 s.
  3. How much work does the pull do in 3 s, and where does it go?
Edit the code in Studio →
Pulling a conducting rod through a magnetic field Scientific model diagram B = 1 T · into the page (top view) B = 1 T · into the page (top view) R = 2 Ω R = 2 Ω bar · 0.5 kg bar · 0.5 kg F = 1 N F = 1 N L = 1 m · Rail length 6 m L = 1 m · Rail length 6 m ε = BLv I = ε/R Fmag = −BLI ε = BLv I = ε/R Fmag = −BLI I > 0: current up the rod; magnetic force opposes motion I > 0: current up the rod; magnetic force opposes motion Smooth horizontal rails; no inductance; x to scale, y schematic Smooth horizontal rails; no inductance; x to scale, y schematic
time 0s distance traveled 0m speed 0m/s induced EMF 0V current 0A magnetic force (+ right) 0N applied work 0J kinetic energy 0J resistive heat 0J Playback speed is adjusted; time readouts still show physical time.
Diagram description

Animated physics diagram: rail_induction; showing time, distance traveled, speed, induced EMF, current, magnetic force (+ right), applied work, kinetic energy, resistive heat.

Unpack one connected process

Speed determines induced EMF; the circuit determines current; current determines magnetic force; that force changes acceleration. This is one feedback process, not a separate motion diagram beside a circuit diagram.

ma = F − B²L²v/R; Wpull = ΔK + Q

1. Which way do the induced current and magnetic force point?

Current flows up the rod and counterclockwise around the loop. The magnetic force points left, opposing the motion. Positive current means up the rod. Reversing the field reverses current, but not the drag.

2. Why does acceleration fade? Find speed and current at 3 s.

ε = BLv and I = BLv/R, so magnetic drag has magnitude B²L²v/R and grows with speed. Initial acceleration is 2 m/s². At 3 s, v ≈ 1.900 m/s and I ≈ 0.950 A. Speed approaches 2 m/s asymptotically; it does not become exactly constant in finite time.

3. How much work does the pull do in 3 s, and where does it go?

Displacement is about 4.100 m, so the pull does 4.100 J of work. Kinetic energy increases by 0.903 J and the resistor dissipates 3.197 J. Their sum equals the applied work. The final rod position is 4.600 m, still on the rails.

Physics reference: OpenStax · Motional EMF

Test your prediction in code

Change the field from 1 T to 2 T. Compare speed and heat at the same time.

scene rail_induction

rail track length=6m separation=1m
rod bar mass=0.5kg length=1m x=0.5m
resistor R 2ohm
field B magnetic 1T direction=into_page
force pull 1N direction=right on bar

bar on track
bar moves through B
track <-> R <-> track

simulate 3s playback=4s
show velocity forces current induction energy heat
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