Sliding Bar on Rails

A conducting bar closes two ideal rails through resistance R and coasts in a uniform transverse magnetic field. Positive x and velocity are rightwards; positive B points into the page. Positive current flows upward through the bar, so EMF(top−bottom)=Bℓv. Reversing B or velocity reverses EMF and current. Fx=−B²ℓ²v/R always opposes velocity. Power I²R=−Fxv is nonnegative.

With no applied driving force, constant mass and a purely resistive circuit: τ=mR/(B²ℓ²), v(t)=v₀exp(−t/τ), Δx=v₀τ[1−exp(−t/τ)]. K=mv²/2, dissipated energy=K₀−K, so their sum remains K₀. Zero B gives τ=∞, constant velocity and Δx=v₀t; initial rest remains rest. Inductance, friction, contacts, heating feedback, radiation and field changes during motion are not modeled.

The controls start a new coast-down when parameters change. Time may be entered precisely; its maximum is the selected plot duration. Automatic duration is 5τ or 5 s without magnetic braking. At 5τ the speed is exp(−5) of its initial value, not exactly zero. Playback moves the timestamp through the full interval in eight real seconds and can pause; it is not a numerical integration or a claim of real-time speed. The schematic horizontal position is scaled to that interval; use numeric Δx for metres.

Field symbols, bar polarity, current, velocity and force directions are signed; arrows are not magnitude scales. B is zero or has |B|≥0.0001 T, bounded by 2 T; ℓ=0.1–1 m, |v₀|≤8 m/s, R=0.05–5 Ω, m=0.05–2 kg. These are arithmetic ranges for an idealized model, not equipment ratings. Drafts/time/window persist through navigation/reload; playback resumes paused. Old five-slider drafts are migrated. Reference: OpenStax motional emf.

Use the compact toolbar for parameters, exact model time and playback. The schematic and velocity plot share the working area; model and playback explanations are available in tooltips. Only entered parameter, time and window drafts are stored. The running playback cursor is temporary: reload restores the last entered time, paused, without a calculated animation snapshot.