Resolve curvature without inventing a neutral gyroperiod
A charged particle moving perpendicular to a uniform stationary magnetic field can change direction without changing speed. For mass m, charge q and field B along +z, define signed rate w=qB/m. Starting at the origin with velocity +1 m/s along x, the supplied path is x=sin(wt)/w and y=[cos(wt)−1]/w for nonzero w. Its initial curvature is toward −y for positive charge and toward +y for negative charge.
Learning outcome: Compare positive, negative and zero charge at the same field. Retain radius/period only when defined, plus exact one-second coordinates. Then double B and predict both radius and period changes before checking.
Starting recap
Use a right-handed coordinate system, oriented surface normal, signed charge/current and declared energy boundary. Separate an entry force from a trajectory, emf from current, and rms from signed average.
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At B=4 mT, q=1 µC and m=2×10⁻⁹ kg, w=2 rad/s, radius .5 m and period π s. After one second, x=sin2/2≈.454649 m and y=(cos2−1)/2≈−.708073 m. Reversing charge reverses y while preserving x and speed.
Open the experimentPredict
At exactly zero charge in this model, the path is:
Experiment
Compare positive, negative and zero charge at the same field. Retain radius/period only when defined, plus exact one-second coordinates. Then double B and predict both radius and period changes before checking.
Resolve magnetic curvature and the zero-charge branch
Worked teaching model · synthetic values · Fictional mass 2e−9 kg enters at origin with vx 1 m/s, vy 0. Uniform stationary field fills the mathematical plane; no electric field, collisions or gravity. Nonrelativistic path shown only over 1 s; charge zero follows a straight line, not an infinite displayed radius.
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Signed path over one second on common spatial scales
Spatial axes use the same pixels per physical unit, preserving geometry. 1: charged orbit segment (solid). Joined curves show the supplied model, not additional observations.
Equivalent numerical coordinates for Signed path over one second on common spatial scales
charged orbit segment
| x (m) | y (m) |
|---|---|
| 0 | 0 |
| 0.0124987 | -0.000156242 |
| 0.0249896 | -0.00062487 |
| 0.0374649 | -0.00140559 |
| 0.0499167 | -0.00249792 |
| 0.0623374 | -0.00390117 |
| 0.0747191 | -0.00561446 |
| 0.0870541 | -0.00763673 |
| 0.0993347 | -0.00996671 |
| 0.111553 | -0.0126029 |
| 0.123702 | -0.0155438 |
| 0.135773 | -0.0187874 |
| 0.14776 | -0.0223318 |
| 0.159654 | -0.0261746 |
| 0.171449 | -0.0303136 |
| 0.183136 | -0.0347462 |
| 0.194709 | -0.0394695 |
| 0.20616 | -0.0444806 |
| 0.217483 | -0.0497764 |
| 0.228669 | -0.0553536 |
| 0.239713 | -0.0612087 |
| 0.250607 | -0.067338 |
| 0.261344 | -0.0737377 |
| 0.271917 | -0.0804038 |
| 0.282321 | -0.0873322 |
| 0.292549 | -0.0945184 |
| 0.302593 | -0.101958 |
| 0.312449 | -0.109647 |
| 0.322109 | -0.117579 |
| 0.331568 | -0.12575 |
| 0.340819 | -0.134156 |
| 0.349858 | -0.142789 |
| 0.358678 | -0.151647 |
| 0.367274 | -0.160722 |
| 0.37564 | -0.170008 |
| 0.383772 | -0.179502 |
| 0.391663 | -0.189195 |
| 0.39931 | -0.199083 |
| 0.406708 | -0.209158 |
| 0.413851 | -0.219416 |
| 0.420735 | -0.229849 |
| 0.427357 | -0.240451 |
| 0.433712 | -0.251214 |
| 0.439795 | -0.262134 |
| 0.445604 | -0.273202 |
| 0.451134 | -0.284412 |
| 0.456382 | -0.295756 |
| 0.461345 | -0.307228 |
| 0.46602 | -0.318821 |
| 0.470403 | -0.330527 |
| 0.474492 | -0.342339 |
| 0.478285 | -0.354249 |
| 0.481779 | -0.366251 |
| 0.484972 | -0.378336 |
| 0.487862 | -0.390497 |
| 0.490447 | -0.402726 |
| 0.492725 | -0.415016 |
| 0.494695 | -0.42736 |
| 0.496356 | -0.439749 |
| 0.497708 | -0.452175 |
| 0.498747 | -0.464631 |
| 0.499476 | -0.47711 |
| 0.499892 | -0.489603 |
| 0.499996 | -0.502102 |
| 0.499787 | -0.5146 |
| 0.499266 | -0.527089 |
| 0.498433 | -0.53956 |
| 0.497288 | -0.552008 |
| 0.495832 | -0.564422 |
| 0.494067 | -0.576797 |
| 0.491993 | -0.589123 |
| 0.489611 | -0.601394 |
| 0.486924 | -0.613601 |
| 0.483932 | -0.625737 |
| 0.480638 | -0.637795 |
| 0.477043 | -0.649767 |
| 0.47315 | -0.661645 |
| 0.468961 | -0.673422 |
| 0.46448 | -0.68509 |
| 0.459708 | -0.696643 |
| 0.454649 | -0.708073 |
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 |
|---|---|---|
| Signed curvature angular rate parameter | 2 | rad/s |
| Particle speed | 1 | m/s |
| Kinetic energy | 1e-9 | J |
| Straight neutral branch | 0 | flag |
| Orbit radius | 0.5 | m |
| Gyroperiod | 3.14159 | s |
| x after one second | 0.454649 | m |
| y after one second | -0.708073 | m |
| Magnetic work over displayed path | 0 | J |
The sign of charge reverses curvature without changing speed or kinetic energy.
With B along +z, positive q curves initially toward −y. Let w=qB/m; x=sin(wt)/w and y=[cos(wt)−1]/w satisfy the initial velocity and Lorentz force. Radius is mv/(|q|B), period 2pi m/(|q|B). Speed independence of this period is restricted to the uniform nonrelativistic model; magnetic force is perpendicular to velocity and does no work.
Predict before inspecting the values. Save contrasting records with assumptions and one alternative explanation. This virtual display does not certify practical measurement technique.
The sign of charge reverses curvature without changing speed or kinetic energy.
With B along +z, positive q curves initially toward −y. Let w=qB/m; x=sin(wt)/w and y=[cos(wt)−1]/w satisfy the initial velocity and Lorentz force. Radius is mv/(|q|B), period 2pi m/(|q|B). Speed independence of this period is restricted to the uniform nonrelativistic model; magnetic force is perpendicular to velocity and does no work.
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