Integrate magnetic rotation and parallel translation
Uniform magnetic force is perpendicular to velocity, so it can turn motion without changing kinetic energy. Use a fictional nonrelativistic particle with mass 1 kg, charge +1 or −1 C, field 1 T along positive z and initial velocity (vperp,0,2) m/s. These parameters define a mathematical experiment, not a real particle or apparatus.
Learning outcome: Compare both charge signs at the same initial velocity, then set transverse speed to zero. Retain position, curvature direction, period, pitch and energy separately. Differentiate the stated position solution to verify that the initial velocity has not silently changed.
Starting recap
Review cross products, source integration measures, line integrals, differential initial states and signed work. Distinguish symmetry premises from integral laws.
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Try
For vperp=3 m/s and positive charge, at t=π/2 s the position is (3,−3,π) m. For negative charge it is (3,+3,π) m, not (−3,+3,π): the initial velocity still points positive x. Both have radius 3 m, period 2π s, pitch 4π m and speed squared 13 m²/s².
Open the experimentPredict
Changing only the charge sign changes which feature?
Experiment
Compare both charge signs at the same initial velocity, then set transverse speed to zero. Retain position, curvature direction, period, pitch and energy separately. Differentiate the stated position solution to verify that the initial velocity has not silently changed.
Integrate the signed magnetic velocity equations
Worked teaching model · synthetic values · Fictional nonrelativistic particle mass 1 kg, charge ±1 C, uniform B=1 T along +z. Initial velocity (vperp,0,2) m/s, initial position zero. No electric field, collisions, radiation or actual particle apparatus. Observe at t=pi/2 s; planar projection uses equal spatial scale.
Reset keeps your prediction, captures and practice history.
Equal-scale projection of the helical path
Spatial axes use the same pixels per physical unit, preserving geometry. 1: projected orbit, longitudinal motion not shown (solid). Joined curves show the supplied model, not additional observations.
Equivalent numerical coordinates for Equal-scale projection of the helical path
projected orbit, longitudinal motion not shown
| x (m) | y (m) |
|---|---|
| 0 | 0 |
| 0.2353773 | -0.009247999 |
| 0.4693034 | -0.03693498 |
| 0.7003361 | -0.08289024 |
| 0.927051 | -0.1468305 |
| 1.14805 | -0.2283614 |
| 1.361971 | -0.3269804 |
| 1.567496 | -0.4420795 |
| 1.763356 | -0.572949 |
| 1.948344 | -0.7187821 |
| 2.12132 | -0.8786797 |
| 2.281218 | -1.051656 |
| 2.427051 | -1.236644 |
| 2.55792 | -1.432504 |
| 2.67302 | -1.638029 |
| 2.771639 | -1.85195 |
| 2.85317 | -2.072949 |
| 2.91711 | -2.299664 |
| 2.963065 | -2.530697 |
| 2.990752 | -2.764623 |
| 3 | -3 |
| 2.990752 | -3.235377 |
| 2.963065 | -3.469303 |
| 2.91711 | -3.700336 |
| 2.85317 | -3.927051 |
| 2.771639 | -4.14805 |
| 2.67302 | -4.361971 |
| 2.55792 | -4.567496 |
| 2.427051 | -4.763356 |
| 2.281218 | -4.948344 |
| 2.12132 | -5.12132 |
| 1.948344 | -5.281218 |
| 1.763356 | -5.427051 |
| 1.567496 | -5.55792 |
| 1.361971 | -5.67302 |
| 1.14805 | -5.771639 |
| 0.927051 | -5.85317 |
| 0.7003361 | -5.91711 |
| 0.4693034 | -5.963065 |
| 0.2353773 | -5.990752 |
| 3.67394e-16 | -6 |
| -0.2353773 | -5.990752 |
| -0.4693034 | -5.963065 |
| -0.7003361 | -5.91711 |
| -0.927051 | -5.85317 |
| -1.14805 | -5.771639 |
| -1.361971 | -5.67302 |
| -1.567496 | -5.55792 |
| -1.763356 | -5.427051 |
| -1.948344 | -5.281218 |
| -2.12132 | -5.12132 |
| -2.281218 | -4.948344 |
| -2.427051 | -4.763356 |
| -2.55792 | -4.567496 |
| -2.67302 | -4.361971 |
| -2.771639 | -4.14805 |
| -2.85317 | -3.927051 |
| -2.91711 | -3.700336 |
| -2.963065 | -3.469303 |
| -2.990752 | -3.235377 |
| -3 | -3 |
| -2.990752 | -2.764623 |
| -2.963065 | -2.530697 |
| -2.91711 | -2.299664 |
| -2.85317 | -2.072949 |
| -2.771639 | -1.85195 |
| -2.67302 | -1.638029 |
| -2.55792 | -1.432504 |
| -2.427051 | -1.236644 |
| -2.281218 | -1.051656 |
| -2.12132 | -0.8786797 |
| -1.948344 | -0.7187821 |
| -1.763356 | -0.572949 |
| -1.567496 | -0.4420795 |
| -1.361971 | -0.3269804 |
| -1.14805 | -0.2283614 |
| -0.927051 | -0.1468305 |
| -0.7003361 | -0.08289024 |
| -0.4693034 | -0.03693498 |
| -0.2353773 | -0.009247999 |
| -7.34788e-16 | 0 |
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.
Independent longitudinal position
Unlike graph quantities rescale independently; read units and values, not screen angles. 1: constant parallel velocity (solid). Joined curves show the supplied model, not additional observations.
Equivalent numerical coordinates for Independent longitudinal position
constant parallel velocity
| time (s) | z (m) |
|---|---|
| 0 | 0 |
| 0.07853982 | 0.1570796 |
| 0.1570796 | 0.3141593 |
| 0.2356194 | 0.4712389 |
| 0.3141593 | 0.6283185 |
| 0.3926991 | 0.7853982 |
| 0.4712389 | 0.9424778 |
| 0.5497787 | 1.099557 |
| 0.6283185 | 1.256637 |
| 0.7068583 | 1.413717 |
| 0.7853982 | 1.570796 |
| 0.863938 | 1.727876 |
| 0.9424778 | 1.884956 |
| 1.021018 | 2.042035 |
| 1.099557 | 2.199115 |
| 1.178097 | 2.356194 |
| 1.256637 | 2.513274 |
| 1.335177 | 2.670354 |
| 1.413717 | 2.827433 |
| 1.492257 | 2.984513 |
| 1.570796 | 3.141593 |
| 1.649336 | 3.298672 |
| 1.727876 | 3.455752 |
| 1.806416 | 3.612832 |
| 1.884956 | 3.769911 |
| 1.963495 | 3.926991 |
| 2.042035 | 4.08407 |
| 2.120575 | 4.24115 |
| 2.199115 | 4.39823 |
| 2.277655 | 4.555309 |
| 2.356194 | 4.712389 |
| 2.434734 | 4.869469 |
| 2.513274 | 5.026548 |
| 2.591814 | 5.183628 |
| 2.670354 | 5.340708 |
| 2.748894 | 5.497787 |
| 2.827433 | 5.654867 |
| 2.905973 | 5.811946 |
| 2.984513 | 5.969026 |
| 3.063053 | 6.126106 |
| 3.141593 | 6.283185 |
| 3.220132 | 6.440265 |
| 3.298672 | 6.597345 |
| 3.377212 | 6.754424 |
| 3.455752 | 6.911504 |
| 3.534292 | 7.068583 |
| 3.612832 | 7.225663 |
| 3.691371 | 7.382743 |
| 3.769911 | 7.539822 |
| 3.848451 | 7.696902 |
| 3.926991 | 7.853982 |
| 4.005531 | 8.011061 |
| 4.08407 | 8.168141 |
| 4.16261 | 8.325221 |
| 4.24115 | 8.4823 |
| 4.31969 | 8.63938 |
| 4.39823 | 8.796459 |
| 4.47677 | 8.953539 |
| 4.555309 | 9.110619 |
| 4.633849 | 9.267698 |
| 4.712389 | 9.424778 |
| 4.790929 | 9.581858 |
| 4.869469 | 9.738937 |
| 4.948008 | 9.896017 |
| 5.026548 | 10.0531 |
| 5.105088 | 10.21018 |
| 5.183628 | 10.36726 |
| 5.262168 | 10.52434 |
| 5.340708 | 10.68142 |
| 5.419247 | 10.83849 |
| 5.497787 | 10.99557 |
| 5.576327 | 11.15265 |
| 5.654867 | 11.30973 |
| 5.733407 | 11.46681 |
| 5.811946 | 11.62389 |
| 5.890486 | 11.78097 |
| 5.969026 | 11.93805 |
| 6.047566 | 12.09513 |
| 6.126106 | 12.25221 |
| 6.204645 | 12.40929 |
| 6.283185 | 12.56637 |
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 cyclotron angular frequency | 1 | rad/s |
| Transverse orbit radius | 3 | m |
| Cyclotron period | 6.283185 | s |
| Pitch along +z | 12.56637 | m |
| Position x at pi over two seconds | 3 | m |
| Position y at pi over two seconds | -3 | m |
| Position z at pi over two seconds | 3.141593 | m |
| Speed squared | 13 | m²/s² |
| Magnetic power | 0 | W |
The perpendicular velocity rotates while the parallel component remains unchanged.
With B along +z, vxprime=(qB/m)vy and vyprime=−(qB/m)vx. For initial vx=vperp, vy=0, vx=vperp cos(omega t), vy=−vperp sin(omega t). Integrating gives x=vperp sin(omega t)/omega and y=vperp[cos(omega t)−1]/omega, with z=2t. Charge sign reverses the transverse rotation but not period or pitch. At zero transverse speed the helix collapses to a straight field-aligned path; q v dot (v cross B)=0 exactly.
Predict before inspecting the values. Save contrasting records with assumptions and one alternative explanation. This virtual display does not certify practical measurement technique.
The perpendicular velocity rotates while the parallel component remains unchanged.
With B along +z, vxprime=(qB/m)vy and vyprime=−(qB/m)vx. For initial vx=vperp, vy=0, vx=vperp cos(omega t), vy=−vperp sin(omega t). Integrating gives x=vperp sin(omega t)/omega and y=vperp[cos(omega t)−1]/omega, with z=2t. Charge sign reverses the transverse rotation but not period or pitch. At zero transverse speed the helix collapses to a straight field-aligned path; q v dot (v cross B)=0 exactly.
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