Distinguish zero net force from stable orientation
A closed loop in a uniform magnetic field can have zero net force and a nonzero torque. For constant B, the integral of I dl crossed with B vanishes because the closed displacement integral is zero. Opposite forces acting at separated parts of the loop can nevertheless form a couple.
Learning outcome: Compare 0°,60°,90°,180° at the same field. Predict the sign of torque from the energy slope before reading the ledger. Explain why zero force, zero torque and a stable energy minimum are three distinct statements.
Starting recap
Review cross products, source integration measures, line integrals, differential initial states and signed work. Distinguish symmetry premises from integral laws.
All lessons are open directly. Your saved experiments remain when you leave and return.
Try
At B=1 T and θ=60°, torque toward increasing angle is −0.3√3 N m≈−0.519615 N m and U=−0.3 J. Slowly rotating from alignment requires external work 0.3 J. Anti-alignment has zero torque but U=+0.6 J, the unstable maximum rather than the stable minimum.
Open the experimentPredict
Why is the anti-aligned zero-torque state unstable in this ideal model?
Experiment
Compare 0°,60°,90°,180° at the same field. Predict the sign of torque from the energy slope before reading the ledger. Explain why zero force, zero torque and a stable energy minimum are three distinct statements.
Distinguish zero net force from a rotational tendency
Worked teaching model · synthetic values · Rigid rectangular loop .5 by .6 m carries maintained current 2 A, magnetic moment .6 A m². Uniform imposed B, angle between moment and B. Torque component is defined positive toward increasing angle. Effective orientation potential U=−m dot B assumes fixed current and imposed field, not the entire source self-energy.
Reset keeps your prediction, captures and practice history.
Effective orientation energy under fixed current
Unlike graph quantities rescale independently; read units and values, not screen angles. 1: uniform imposed field (solid). Joined curves show the supplied model, not additional observations.
Equivalent numerical coordinates for Effective orientation energy under fixed current
uniform imposed field
| moment angle (degrees) | orientation energy (J) |
|---|---|
| 0 | -0.6 |
| 2.25 | -0.5995374 |
| 4.5 | -0.5981504 |
| 6.75 | -0.5958411 |
| 9 | -0.592613 |
| 11.25 | -0.5884712 |
| 13.5 | -0.583422 |
| 15.75 | -0.5774731 |
| 18 | -0.5706339 |
| 20.25 | -0.5629148 |
| 22.5 | -0.5543277 |
| 24.75 | -0.5448859 |
| 27 | -0.5346039 |
| 29.25 | -0.5234976 |
| 31.5 | -0.5115841 |
| 33.75 | -0.4988818 |
| 36 | -0.4854102 |
| 38.25 | -0.4711902 |
| 40.5 | -0.4562436 |
| 42.75 | -0.4405935 |
| 45 | -0.4242641 |
| 47.25 | -0.4072804 |
| 49.5 | -0.3896688 |
| 51.75 | -0.3714564 |
| 54 | -0.3526712 |
| 56.25 | -0.3333421 |
| 58.5 | -0.3134991 |
| 60.75 | -0.2931727 |
| 63 | -0.2723943 |
| 65.25 | -0.2511958 |
| 67.5 | -0.2296101 |
| 69.75 | -0.2076702 |
| 72 | -0.1854102 |
| 74.25 | -0.1628643 |
| 76.5 | -0.1400672 |
| 78.75 | -0.1170542 |
| 81 | -0.09386068 |
| 83.25 | -0.07052244 |
| 85.5 | -0.04707546 |
| 87.75 | -0.02355589 |
| 90 | -3.67394e-17 |
| 92.25 | 0.02355589 |
| 94.5 | 0.04707546 |
| 96.75 | 0.07052244 |
| 99 | 0.09386068 |
| 101.25 | 0.1170542 |
| 103.5 | 0.1400672 |
| 105.75 | 0.1628643 |
| 108 | 0.1854102 |
| 110.25 | 0.2076702 |
| 112.5 | 0.2296101 |
| 114.75 | 0.2511958 |
| 117 | 0.2723943 |
| 119.25 | 0.2931727 |
| 121.5 | 0.3134991 |
| 123.75 | 0.3333421 |
| 126 | 0.3526712 |
| 128.25 | 0.3714564 |
| 130.5 | 0.3896688 |
| 132.75 | 0.4072804 |
| 135 | 0.4242641 |
| 137.25 | 0.4405935 |
| 139.5 | 0.4562436 |
| 141.75 | 0.4711902 |
| 144 | 0.4854102 |
| 146.25 | 0.4988818 |
| 148.5 | 0.5115841 |
| 150.75 | 0.5234976 |
| 153 | 0.5346039 |
| 155.25 | 0.5448859 |
| 157.5 | 0.5543277 |
| 159.75 | 0.5629148 |
| 162 | 0.5706339 |
| 164.25 | 0.5774731 |
| 166.5 | 0.583422 |
| 168.75 | 0.5884712 |
| 171 | 0.592613 |
| 173.25 | 0.5958411 |
| 175.5 | 0.5981504 |
| 177.75 | 0.5995374 |
| 180 | 0.6 |
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 |
|---|---|---|
| Loop magnetic moment magnitude | 0.6 | A m² |
| Torque toward increasing angle | -0.5196152 | N m |
| Effective orientation energy | -0.3 | J |
| External quasistatic work from aligned state | 0.3 | J |
| Net force in uniform field | 0 | N |
| Angular curvature of effective energy | 0.3 | J/rad² |
A uniform field gives zero net force on the closed loop but can produce a nonzero torque.
Integrate I dl cross B around a closed loop: constant B factors out and the closed displacement integral is zero. Opposite forces at separated sides still form a couple, giving tau=m cross B with m=I A n. For the declared increasing-angle coordinate tau_theta=−mB sin(theta)=−dU/dtheta and U=−mB cos(theta). Both aligned and anti-aligned orientations have zero torque, but their energy curvatures have opposite signs. This effective fixed-current interaction energy is not an accounting of all circuit-source self-energy.
Predict before inspecting the values. Save contrasting records with assumptions and one alternative explanation. This virtual display does not certify practical measurement technique.
A uniform field gives zero net force on the closed loop but can produce a nonzero torque.
Integrate I dl cross B around a closed loop: constant B factors out and the closed displacement integral is zero. Opposite forces at separated sides still form a couple, giving tau=m cross B with m=I A n. For the declared increasing-angle coordinate tau_theta=−mB sin(theta)=−dU/dtheta and U=−mB cos(theta). Both aligned and anti-aligned orientations have zero torque, but their energy curvatures have opposite signs. This effective fixed-current interaction energy is not an accounting of all circuit-source self-energy.
Captures preserve recipes and replay their calculated results. They are synthetic model evidence, not physical measurements.
Understand
Apply
Public practice checks a specific claim. It does not establish independent mastery.
Which source geometry, symmetry, initial state or calibration premise is necessary, and what independent observation could reject it?
Continue
Completion requires a saved prediction, at least one capture and a reflection. Presence checks do not grade your reasoning.
Skipping retains experiments and awards no mastery.
Save a science backup or import one
Science work is stored locally in this browser, separately from mathematics and AI. It does not sync automatically. Use this science-only backup; this checkout's older all-subject export does not include science.
A legacy file is available only when the complete snapshot fits the 36 MB limit. For larger snapshots, prepare a complete file set below; never discard notebooks to make a smaller file.
Matching imported notebook drafts replace their current drafts. Omitted lessons and known first attempts remain. Export your current work first if you want to retain both drafts.
Complete file sets and course backups
A file set can hold more work than one science file. Keep its manifest and every part together. Each file is limited to 36 MB. This format needs this version of LearnAI or a compatible newer version; older versions can use only the legacy single-file backup above.
The set includes every saved science course and global continuation at this moment. Later edits need a new export. A conflicting tab exports its own visible work, which may differ from saved work in another tab.
0 file(s) selected. File order and filenames do not determine identity. Missing, changed or conflicting parts block the entire restore; no good part is applied by itself.
Science storage: loading. No account or remote computation is required.