Magnetic Force & Lorentz Force

Step-by-step

1. The magnetic field B points along the z-axis (blue arrow).
2. The charged particle enters moving in the x-direction.
3. Observe the circular path.
4. Flip the charge sign to reverse the rotation direction.
5. Add a z-velocity component (Pro) to see helical motion.
6. Watch how increasing B tightens the circular orbit.

Key formulas

• $\vec{F} = q(\vec{v} \times \vec{B})$Lorentz force (magnetic component)
• $F = qvB\sin\theta$Magnitude of magnetic force
• $r = \frac{mv}{|q|B}$Radius of circular motion (cyclotron radius)
• $T = \frac{2\pi m}{|q|B}$Period of circular motion (independent of speed)
• $\omega_c = \frac{|q|B}{m}$Cyclotron frequency

Frequently asked questions

A proton (m=1.67×10⁻²⁷ kg, q=1.6×10⁻¹⁹ C) moves at 2×10⁶ m/s in a 0.5 T field. Find the radius of its circular path.

R = mv/(qB).

Why does the magnetic force do no work on the particle?

You can work it out this way: work = F·d. The force is always perpendicular to displacement.

If you double the magnetic field strength, how does the radius change?

R = mv/(qB) — radius is inversely proportional to B.

An electron enters a 0.2 T field at 3×10⁶ m/s at 30° to the field direction. Describe the resulting path.

Decompose v into parallel and perpendicular components. Parallel → no force, perpendicular → circular. Combined = helix.

A cyclotron accelerates protons with B = 1.5 T. What is the period of circulation? Is it speed-dependent?

T = 2πm/(qB). The period is independent of speed — that's what makes cyclotrons work.