A charged particle moving through a magnetic field experiences a force perpendicular to its velocity (the Lorentz force) and undergoes circular motion.
Basic Knowledge
Lorentz force:F = qvB sinθ (v: speed, θ: angle between velocity and magnetic field)
Direction: right-hand rule (for a positive charge: the direction of v × B) or Fleming’s left-hand rule (pay attention to the sign of the charge)
The Lorentz force is perpendicular to velocity → does no work → speed does not change.
Circular motion of a charged particle entering perpendicular to a magnetic field:
Lorentz force = centripetal force:qvB = mv²/r → r = mv/(qB) (cyclotron radius)
Period: T = 2πm/(qB) (independent of speed!)
Frequency: f = qB/(2πm) (cyclotron frequency)
When electric and magnetic fields coexist:F = qE + qv × B (general formula for the Lorentz force)
Lorentz force (Lorentz force) (Force acting on a moving charge in a magnetic field. F = qvB sinθ)
Cyclotron radius r (r = mv/(qB). It increases as mass or speed increases)
Cyclotron frequency (f = qB/(2πm). Constant regardless of speed)
Hall effect (The phenomenon in which a voltage develops in a conductor due to the interaction of current and a magnetic field)
Velocity selector (A device that applies electric and magnetic fields in opposite directions and allows particles of a specific speed to travel straight)
Further Study (Background and Significance)
A cyclotron, a type of particle accelerator, uses this principle. The particle’s speed increases each time it is accelerated through a semicircle, but because the period T = 2πm/(qB) does not depend on speed, it can continue to be accelerated by an alternating voltage of the same frequency (when relativistic effects can be ignored at low speed).
The aurora is also a result of the Lorentz force: cosmic rays (charged particles) are captured by Earth’s magnetic field and move helically, colliding with atmospheric molecules and emitting light.
The Hall effect is widely used in practice as a magnetic-field sensor (Hall element), including in automobile engine control systems and current meters.
- F = qvB sinθ (perpendicular to velocity, zero work)
- Perpendicular entry into a magnetic field → circular motion
- r = mv/(qB) (cyclotron radius)
- T = 2πm/(qB) (period independent of speed)
- The Lorentz-force direction is opposite for positive and negative charges
- Check the direction using Fleming’s left-hand rule
- Coexisting electric and magnetic fields: F = qE + qv×B
Points to Note (Easily Confused)
① The Lorentz force does no work (speed remains constant). ② The Lorentz-force direction is opposite for positive and negative charges. ③ The velocity component parallel to the magnetic field experiences no Lorentz force (helical motion). ④ It is important that the cyclotron frequency is “independent of speed.”
Practice
- Find the radius of the circular motion of an electron entering perpendicular to a magnetic field with flux density B = 0.20 T at speed v = 3.0 × 10⁶ m/s (m = 9.1 × 10⁻³¹ kg, q = 1.6 × 10⁻¹⁹ C).
- Explain why the Lorentz force does no work.
- When a proton and an electron move at the same speed in the same magnetic field, find the ratio of their circular-motion radii (assume that the proton’s mass is 1836 times the electron’s mass).