Force on Moving Charges
A short lesson with diagrams, a worked example, a problem to try, and a quick check. Then quiz yourself on 6 practice questions.
A magnetic field pushes on a charge only if the charge is moving across the field: F = qvB sin θ. The force is always perpendicular to both v and B, so it changes the direction of motion but never the speed. In a uniform field, a charge moving perpendicular to B goes in a circle.
1. Size and direction
2. The right-hand rule
Point your fingers along v, curl them toward B; your thumb gives the force on a positive charge. (Or: fingers along B, thumb along v, palm pushes in the direction of F.)
For an electron, flip the answer.
Because F ⊥ v, the magnetic force does no work. Kinetic energy and speed stay constant.
Setting qvB = mv²/r gives the radius: r = mv/(|q|B). Faster or heavier particles make bigger circles; a stronger field makes tighter ones.
3. Example, then try it
A proton moves at 3.0 × 10⁶ m/s perpendicular to a 0.20 T field. Find the magnetic force on it.
F = qvB = (1.6 × 10⁻¹⁹)(3.0 × 10⁶)(0.20) = 9.6 × 10⁻¹⁴ N.
That proton (m = 1.67 × 10⁻²⁷ kg) circles in the field. Find the radius of its circle. How would the radius change if its speed doubled? What would an electron at the same speed do?
Radius: r = mv/(qB) = (1.67 × 10⁻²⁷)(3.0 × 10⁶) ÷ [(1.6 × 10⁻¹⁹)(0.20)] ≈ 0.16 m.
Double the speed: r ∝ v, so the radius doubles to about 0.31 m.
Electron: it curves the opposite way, in a much smaller circle (its mass is about 1/1800 of the proton’s): r ≈ 8.5 × 10⁻⁵ m.
Answer: about 0.16 m; it doubles; an electron curls the other way in a tiny circle.
4. Common mistakes
How much work does a magnetic field do on a charged particle moving in a circle?
The magnetic force is always perpendicular to the velocity, so it does no work. The particle’s speed and kinetic energy stay constant.
Ready to practice?
6 questions on Force on Moving Charges, each with a hint and an explanation for every wrong answer.