Separate magnetic guiding drift from electric work
A particle released from rest in crossed uniform fields is not the same experiment as a force-free velocity selector. Here a fictional mass 1 kg and charge ±1 C starts at the origin, with B=1 T along positive z and E along positive x. The electric field initially accelerates it; magnetic force then bends the growing motion.
Learning outcome: Compare charge signs and then compare E=0 with nonzero E. Retain electric work and guiding drift rather than attributing the energy gain to magnetic curvature. Substitute the force-free selector velocity into the equations and explain why it differs from the displayed initial state.
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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With E=2 V/m and positive charge at t=π s, position is (4,−2π) m and velocity is (0,−4) m/s. Kinetic energy has increased by 8 J, equal to qEx=1×2×4. Negative charge instead gives x=−4 m and the same y and kinetic energy; its negative charge times negative displacement still gives positive electric work.
Open the experimentPredict
Which field supplies the rest-start kinetic-energy increase?
Experiment
Compare charge signs and then compare E=0 with nonzero E. Retain electric work and guiding drift rather than attributing the energy gain to magnetic curvature. Substitute the force-free selector velocity into the equations and explain why it differs from the displayed initial state.
Solve electric work and magnetic guiding drift
Worked teaching model · synthetic values · Fictional mass 1 kg and charge ±1 C start from rest at origin; uniform B=1 T along +z and uniform E along +x. Nonrelativistic prescribed fields, no collisions or radiation. Observe t=pi s and plot through one cyclotron period. No physical beam or crossed-field apparatus.
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Electric work with a charge-independent guiding drift
Spatial axes use the same pixels per physical unit, preserving geometry. 1: trajectory from rest (solid). Joined curves show the supplied model, not additional observations.
Equivalent numerical coordinates for Electric work with a charge-independent guiding drift
trajectory from rest
| x (m) | y (m) |
|---|---|
| 0 | 0 |
| 0.006165333 | -0.0001614412 |
| 0.02462332 | -0.001290335 |
| 0.05526016 | -0.00434817 |
| 0.09788697 | -0.01028454 |
| 0.1522409 | -0.0200313 |
| 0.217987 | -0.0344968 |
| 0.2947197 | -0.0545603 |
| 0.381966 | -0.08106656 |
| 0.4791881 | -0.1148206 |
| 0.5857864 | -0.1565828 |
| 0.7011039 | -0.207064 |
| 0.8244295 | -0.2669216 |
| 0.9550029 | -0.3367549 |
| 1.092019 | -0.4171018 |
| 1.234633 | -0.5084354 |
| 1.381966 | -0.6111611 |
| 1.533109 | -0.7256139 |
| 1.687131 | -0.8520567 |
| 1.843082 | -0.9906784 |
| 2 | -1.141593 |
| 2.156918 | -1.304838 |
| 2.312869 | -1.480375 |
| 2.466891 | -1.668092 |
| 2.618034 | -1.867798 |
| 2.765367 | -2.079232 |
| 2.907981 | -2.302057 |
| 3.044997 | -2.53587 |
| 3.175571 | -2.780196 |
| 3.298896 | -3.034497 |
| 3.414214 | -3.298175 |
| 3.520812 | -3.570573 |
| 3.618034 | -3.850978 |
| 3.70528 | -4.138631 |
| 3.782013 | -4.432727 |
| 3.847759 | -4.73242 |
| 3.902113 | -5.036833 |
| 3.94474 | -5.345056 |
| 3.975377 | -5.656157 |
| 3.993835 | -5.969187 |
| 4 | -6.283185 |
| 3.993835 | -6.597183 |
| 3.975377 | -6.910214 |
| 3.94474 | -7.221315 |
| 3.902113 | -7.529538 |
| 3.847759 | -7.83395 |
| 3.782013 | -8.133644 |
| 3.70528 | -8.42774 |
| 3.618034 | -8.715393 |
| 3.520812 | -8.995798 |
| 3.414214 | -9.268195 |
| 3.298896 | -9.531873 |
| 3.175571 | -9.786175 |
| 3.044997 | -10.0305 |
| 2.907981 | -10.26431 |
| 2.765367 | -10.48714 |
| 2.618034 | -10.69857 |
| 2.466891 | -10.89828 |
| 2.312869 | -11.086 |
| 2.156918 | -11.26153 |
| 2 | -11.42478 |
| 1.843082 | -11.57569 |
| 1.687131 | -11.71431 |
| 1.533109 | -11.84076 |
| 1.381966 | -11.95521 |
| 1.234633 | -12.05794 |
| 1.092019 | -12.14927 |
| 0.9550029 | -12.22962 |
| 0.8244295 | -12.29945 |
| 0.7011039 | -12.35931 |
| 0.5857864 | -12.40979 |
| 0.4791881 | -12.45155 |
| 0.381966 | -12.4853 |
| 0.2947197 | -12.51181 |
| 0.217987 | -12.53187 |
| 0.1522409 | -12.54634 |
| 0.09788697 | -12.55609 |
| 0.05526016 | -12.56202 |
| 0.02462332 | -12.56508 |
| 0.006165333 | -12.56621 |
| 0 | -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 |
|---|---|---|
| Position x at pi seconds | 4 | m |
| Position y at pi seconds | -6.283185 | m |
| Velocity x at pi seconds | 0 | m/s |
| Velocity y at pi seconds | -4 | m/s |
| Electric work | 8 | J |
| Kinetic energy increase | 8 | J |
| Work-energy residual | 0 | J |
| Guiding drift along y | -2 | m/s |
| Magnetic power | 0 | W |
The magnetic field changes the trajectory but the electric field supplies the kinetic-energy change.
For omega=qB/m and vD=E/B, the rest-start solution is vx=vD sin(omega t), vy=vD[cos(omega t)−1]. The guiding drift is E cross B divided by B², here along −y, independent of charge sign. Integrating gives x=vD[1−cos(omega t)]/omega and y=vD[sin(omega t)/omega−t]. The electric work qE x equals kinetic-energy increase; magnetic power remains zero. A force-free selector requires the special initial velocity (0,−E/B), not the rest-start state used here.
Predict before inspecting the values. Save contrasting records with assumptions and one alternative explanation. This virtual display does not certify practical measurement technique.
The magnetic field changes the trajectory but the electric field supplies the kinetic-energy change.
For omega=qB/m and vD=E/B, the rest-start solution is vx=vD sin(omega t), vy=vD[cos(omega t)−1]. The guiding drift is E cross B divided by B², here along −y, independent of charge sign. Integrating gives x=vD[1−cos(omega t)]/omega and y=vD[sin(omega t)/omega−t]. The electric work qE x equals kinetic-energy increase; magnetic power remains zero. A force-free selector requires the special initial velocity (0,−E/B), not the rest-start state used here.
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