Magnetic Force on 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 moving charge with F = qv × B. The force is always perpendicular to the velocity, so it changes direction, not speed: a charge moving across a uniform field goes in a circle of radius r = mv/(qB).
1. The force and the circle
2. What the motion looks like
Faster charges make bigger circles but take the same time per lap: r ∝ v, so the circumference grows as fast as the speed. That’s why a cyclotron works at one fixed frequency.
Velocity at an angle to B: the component along B is unaffected, so the path is a helix.
Mass spectrometer: accelerate ions through ΔV (qΔV = ½mv²), then bend them in B. Heavier ions make wider circles: r = (1/B)√(2mΔV/q).
Crossed E and B: only charges with v = E/B pass straight through, whatever their charge or mass.
3. Example, then try it
A proton (m = 1.67 × 10⁻²⁷ kg) moves at 2.0 × 10⁶ m/s perpendicular to a 0.50 T field. Find the radius of its path and the time for one lap.
r = mv/(qB) = (1.67 × 10⁻²⁷)(2.0 × 10⁶) ÷ [(1.6 × 10⁻¹⁹)(0.50)] ≈ 4.2 cm.
T = 2πm/(qB) = 2π(1.67 × 10⁻²⁷) ÷ (8.0 × 10⁻²⁰) ≈ 1.3 × 10⁻⁷ s.
In a mass spectrometer, singly charged carbon-12 ions (m = 2.0 × 10⁻²⁶ kg) start from rest, are accelerated through 1000 V, then enter a 0.40 T field at right angles. Find their speed, the radius of their path, and where they land relative to the entry point.
Speed: qΔV = ½mv² → v = √(2qΔV/m) = √[2(1.6 × 10⁻¹⁹)(1000) ÷ (2.0 × 10⁻²⁶)] ≈ 1.3 × 10⁵ m/s.
Radius: r = mv/(qB) = (2.0 × 10⁻²⁶)(1.26 × 10⁵) ÷ [(1.6 × 10⁻¹⁹)(0.40)] ≈ 3.9 cm.
Landing spot: after a half circle, one diameter away: about 7.9 cm. A heavier isotope lands farther out.
Answer: about 1.3 × 10⁵ m/s; r ≈ 3.9 cm; it lands about 7.9 cm from the entrance.
4. Common mistakes
A charged particle circles in a uniform magnetic field. The field strength is doubled (same speed). What happens to the radius?
r = mv/(qB), so doubling B halves r.
Ready to practice?
6 questions on Magnetic Force on Charges, each with a hint and an explanation for every wrong answer.