Circles & Helices
Bent into circles, never sped up.
Lesson at a glanceShow the key points
- Magnetic force does no work: speed is constant.
- v ⊥ B: r = mv/qB (34.2), T = 2πm/qB (34.3), ν = qB/2πm (34.4).
- Period independent of speed.
- Oblique velocity: helix, r = mv⊥/qB, pitch = v∥T.
Because the magnetic force always acts sideways, it can bend a charge’s path but never speed it up. In a uniform field the result is a circle, or a helix.
No work, constant speed
The magnetic force is perpendicular to the velocity, so it does no work. The particle’s kinetic energy, and so its speed, never changes.
Diagrams show a field into the page as ⊗ (the tail of an arrow) and out of the page as ⊙ (its tip). Let a charge q move at speed v perpendicular to a uniform field. The force qvB is constant in size and always perpendicular to the motion, so it acts as a centripetal force:
The period does not depend on the speed: a faster particle runs a bigger circle in the same time. The frequency ν is called the cyclotron frequency.
A particle of charge 50 µC and mass 5.0 mg is projected at 10 m/s perpendicular to a uniform field of 40 mT. When does it first return to its starting point?
Singly charged neon-20 ions with kinetic energy 1.5 keV enter a 0.50 T field at right angles. Find the radius of their path, and the radius for neon-22 ions of the same energy. (Take the ion mass as A × 1.67 × 10⁻²⁷ kg.)
Velocity at an angle: a helix
If v⃗ isn’t perpendicular to B⃗, split it into v∥ along the field and v⊥ across it. The force is perpendicular to B⃗, so v∥ never changes; meanwhile the particle circles with radius r = mv⊥/qB. Together: a helix wound round the field lines. In one period it advances a distance called the pitch, v∥T.
Protons at 5.0 × 10⁵ m/s enter a 0.40 T field at 30° to the field direction. Find the radius and pitch of their helix.
A particle of charge 10 µC and mass 10 µg circles with radius 4.0 cm in a 0.50 T field. At one point an electric field is switched on, after which it moves in a straight line at constant speed. Find E.
- 1The magnetic force does no work: speed is constant.
- 2v ⊥ B: circle with r = mv/qB (34.2), T = 2πm/qB (34.3), ν = qB/2πm (34.4), independent of speed.
- 3General angle: helix with r = mv⊥/qB and pitch v∥T.
Check yourself
Keep exploring
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