Physics HL · Chapter 19: Motion in Electric and Magnetic Fields
19.3 Magnetic Work, Helical Motion, and Spiral Paths
Prove why magnetic forces do zero work, then model mixed-component motion and the conditions that produce spiral radius change.
Estimated time: 34 minutes
Why Magnetic Force Does Zero Work
Mechanical work is F dot s. In magnetic motion, the force direction is always perpendicular to instantaneous displacement direction, so their dot product is zero. That means a magnetic field alone cannot change kinetic energy. If speed changes in a setup that includes B, some other force (usually electric) must be transferring energy.
Zero work does not mean zero force; it means force is purely directional, not speed-changing.
This result explains a common observation: a particle can leave a magnetic region with the same speed it had on entry but with a different direction. The momentum vector changed, but the kinetic-energy scalar did not.
Helical Motion from Parallel and Perpendicular Components
If velocity has a component parallel to B and a component perpendicular to B, the perpendicular component drives circular motion while the parallel component remains unchanged. Superposing these produces a helix. Radius is set by v_perpendicular, while pitch is set by v_parallel times period.
r = rac{mv_perp}{|q|B},qquad ext{pitch} = v_parallel T = v_parallel rac{2pi m}{|q|B}
Increasing launch angle toward 90 degrees increases radius and decreases pitch.
Because the parallel component is unaffected by magnetic force, helix pitch stays constant in a uniform field when no other forces act. This is why many beam-transport devices separate trajectory geometry into independent transverse and longitudinal design problems.
When and Why Spiral Paths Appear
A true spiral with changing radius requires speed change or magnetic-field change. For example, if a particle loses kinetic energy in matter while still in a field, v_perpendicular decreases and radius shrinks inward. If field magnitude increases along the path, radius can also contract. Spiral behavior therefore indicates that at least one assumption of ideal uniform B-only motion has been broken.
Simulation: Helix Pitch and Radius Control
Use launch-angle decomposition to control v_parallel and v_perpendicular independently, then read off helix pitch and turn radius.
Path class
Helix
Radius
9.78e-4 m
Period
5.95e-10 s
Frequency
1.68e+9 Hz
Helix pitch
8.78e-3 m
Magnetic force magnitude
9.92e-14 N
v perpendicular
1.03e+7 m/s
v parallel
1.47e+7 m/s
Test Yourself
Which statement is always true for ideal motion in a uniform magnetic field with no electric field?