Close a finite-mass motional-induction energy account
A sliding rod can induce a current that opposes the mechanical drive, but its speed need not instantly equal a terminal value. In this original ideal model mass, rod length, field magnitude and total resistance are each one in SI units. A constant force F drives it from rest. Friction, loop inductance and radiation are neglected explicitly.
Learning outcome: Compare early and late times at fixed drive, then change the drive force. Retain velocity, displacement, kinetic storage and heat. Explain what additional state would be needed if loop inductance were no longer negligible.
Starting recap
Review flux integrals, linear differential equations, initial conditions, complex numbers and vector derivatives. Keep source constraints, RMS conventions and calibration roles explicit.
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With F=2 N at t=1 s, speed is 2(1−1/e)≈1.264241 m/s and displacement 2/e≈0.735759 m. Drive work is about 1.471518 J. Kinetic energy is about 0.799153 J and resistive conversion about 0.672365 J; the totals close without assuming terminal motion.
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Why is current proportional to instantaneous speed in this particular model?
Experiment
Compare early and late times at fixed drive, then change the drive force. Retain velocity, displacement, kinetic storage and heat. Explain what additional state would be needed if loop inductance were no longer negligible.
Solve a driven rod with magnetic damping
Worked teaching model · synthetic values · Ideal sliding rod mass1 kg,length1 m,B1 T,total resistance1 ohm, starting from rest. Neglect loop inductance, friction and radiation; mechanical drive is constant. Induced current gives force −v in SI, so vprime=F−v. All motion is virtual; no current apparatus.
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Finite-mass approach to magnetic terminal speed
Unlike graph quantities rescale independently; read units and values, not screen angles. 1: solution of vprime=F−v (solid). Joined curves show the supplied model, not additional observations.
Equivalent numerical coordinates for Finite-mass approach to magnetic terminal speed
solution of vprime=F−v
| time (s) | rod speed (m/s) |
|---|---|
| 0 | 0 |
| 0.05 | 0.09754115 |
| 0.1 | 0.1903252 |
| 0.15 | 0.278584 |
| 0.2 | 0.3625385 |
| 0.25 | 0.4423984 |
| 0.3 | 0.5183636 |
| 0.35 | 0.5906238 |
| 0.4 | 0.6593599 |
| 0.45 | 0.7247437 |
| 0.5 | 0.7869387 |
| 0.55 | 0.8461004 |
| 0.6 | 0.9023767 |
| 0.65 | 0.9559084 |
| 0.7 | 1.006829 |
| 0.75 | 1.055267 |
| 0.8 | 1.101342 |
| 0.85 | 1.14517 |
| 0.9 | 1.186861 |
| 0.95 | 1.226518 |
| 1 | 1.264241 |
| 1.05 | 1.300125 |
| 1.1 | 1.334258 |
| 1.15 | 1.366726 |
| 1.2 | 1.397612 |
| 1.25 | 1.42699 |
| 1.3 | 1.454936 |
| 1.35 | 1.481519 |
| 1.4 | 1.506806 |
| 1.45 | 1.530859 |
| 1.5 | 1.55374 |
| 1.55 | 1.575504 |
| 1.6 | 1.596207 |
| 1.65 | 1.6159 |
| 1.7 | 1.634633 |
| 1.75 | 1.652452 |
| 1.8 | 1.669402 |
| 1.85 | 1.685526 |
| 1.9 | 1.700863 |
| 1.95 | 1.715452 |
| 2 | 1.729329 |
| 2.05 | 1.74253 |
| 2.1 | 1.755087 |
| 2.15 | 1.767032 |
| 2.2 | 1.778394 |
| 2.25 | 1.789202 |
| 2.3 | 1.799482 |
| 2.35 | 1.809262 |
| 2.4 | 1.818564 |
| 2.45 | 1.827413 |
| 2.5 | 1.83583 |
| 2.55 | 1.843837 |
| 2.6 | 1.851453 |
| 2.65 | 1.858698 |
| 2.7 | 1.865589 |
| 2.75 | 1.872144 |
| 2.8 | 1.87838 |
| 2.85 | 1.884311 |
| 2.9 | 1.889954 |
| 2.95 | 1.895321 |
| 3 | 1.900426 |
| 3.05 | 1.905282 |
| 3.1 | 1.909902 |
| 3.15 | 1.914296 |
| 3.2 | 1.918476 |
| 3.25 | 1.922452 |
| 3.3 | 1.926234 |
| 3.35 | 1.929831 |
| 3.4 | 1.933253 |
| 3.45 | 1.936509 |
| 3.5 | 1.939605 |
| 3.55 | 1.942551 |
| 3.6 | 1.945353 |
| 3.65 | 1.948018 |
| 3.7 | 1.950553 |
| 3.75 | 1.952965 |
| 3.8 | 1.955258 |
| 3.85 | 1.957441 |
| 3.9 | 1.959516 |
| 3.95 | 1.961491 |
| 4 | 1.963369 |
At most 82 representative computed coordinates including endpoints; the plotted curve can use a finer grid. A table of a revealed generator is not independent measurement evidence.
| Quantity | Value | Unit |
|---|---|---|
| Rod speed | 1.264241 | m/s |
| Rod displacement | 0.7357589 | m |
| Motional emf magnitude | 1.264241 | V |
| Induced current magnitude | 1.264241 | A |
| Magnetic drag magnitude | 1.264241 | N |
| Drive work | 1.471518 | J |
| Kinetic energy | 0.7991528 | J |
| Resistor conversion | 0.672365 | J |
| Energy residual | 0 | J |
| Terminal speed | 2 | m/s |
External work divides between kinetic storage and resistive conversion; magnetic drag is not a free energy source.
The moving rod gives emf=B ell v and I=emf/R; its magnetic force opposes motion with magnitude B²ell²v/R. With the stated constants, vprime=F−v and v=F(1−e^(−t)). Integrate v to obtain displacement and integrate I²R to obtain heat. The no-inductance approximation makes current instantaneous; a real loop with appreciable inductance would add a dynamic state and magnetic storage.
Predict before inspecting the values. Save contrasting records with assumptions and one alternative explanation. This virtual display does not certify practical measurement technique.
External work divides between kinetic storage and resistive conversion; magnetic drag is not a free energy source.
The moving rod gives emf=B ell v and I=emf/R; its magnetic force opposes motion with magnitude B²ell²v/R. With the stated constants, vprime=F−v and v=F(1−e^(−t)). Integrate v to obtain displacement and integrate I²R to obtain heat. The no-inductance approximation makes current instantaneous; a real loop with appreciable inductance would add a dynamic state and magnetic storage.
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