Lesson 7.2 • Competency 7.1Complete Lesson

Forces on Moving Charges & Hall Effect

Lorentz magnetic force F = q(v × B), circular and helical trajectories in uniform B-fields, cyclotron radius r = mv/(qB) and frequency, velocity selector v = E/B, J.J. Thomson’s e/m experiment, and the Hall effect V_H = BI/(nqt).

Marking a lesson complete represents finishing your study of the topic; it does not denote tested mastery.

Learning Objectives

Calculate the magnetic deflecting force on moving charges using F = B q v sin θ, using Fleming’s Left-Hand Rule or vector cross-products.
Derive the radius r = mv/(qB) and period T = 2πm/(qB) for circular cyclotron motion, explaining why magnetic forces do zero mechanical work.
Explain the principle of the velocity selector (v = E/B) and the Hall effect, deriving Hall voltage V_H = BI/(nqt) and identifying majority charge carriers.

Core Theory & Concepts

Stationary charges experience no force in a pure magnetic field. However, once in motion, a charged particle interacts with the magnetic flux, experiencing a perpendicular deflecting Lorentz force. This principle powers mass spectrometers, particle cyclotrons, auroras, and modern Hall sensors.

Lorentz Magnetic Force

The magnetic force acts perpendicular to both velocity and field: F = |q| v B sin θ. Because F is always perpendicular to velocity, instantaneous work done is zero.

\vec{F} = q(\vec{v} \times \vec{B}), \quad F = |q| v B \sin\theta

Cyclotron Orbit & Radius

Centripetal force is provided entirely by the magnetic force: m v² / r = q v B => r = (m v) / (q B). The orbital period T = 2πm / (qB) is notably independent of speed.

r = \frac{m v}{q B}, \quad T = \frac{2\pi m}{q B}

Hall Voltage

In a current-carrying slab in a perpendicular field, magnetic deflection separates carriers until transverse electric force balances magnetic force: V_H = (B I) / (n q t).

V_H = \frac{B I}{n q t}

Common Misconceptions

Misconception

A magnetic field can accelerate a charged particle to increase its kinetic energy and speed.

Physical Clarification:A steady magnetic field can only alter the direction of velocity (centripetal acceleration), never its speed or magnitude. Because F ⊥ v, the work done is strictly zero.

Syllabus Topic Outline

Competency 7.1

1Magnetic force on moving charge: F = q(v × B) and Fleming’s Left-Hand Rule
2Proof that magnetic forces do no work (F ⊥ v ⇒ P = F · v = 0)
3Circular motion in perpendicular B-fields: radius, period, and cyclotron frequency
4Helical trajectories for oblique entry velocity components
5Velocity selector (crossed E and B fields) and Thomson e/m measurement
6Hall effect: derivation of V_H = BI / (nqt) and carrier sign determination