Lorentz Force

The Lorentz force law is the single most general expression describing the force experienced by a charged particle in the presence of both electric and magnetic fields. It unifies electrostatic and magnetostatic interactions into one compact expression and serves as the foundation for understanding charged particle motion in accelerators, cathode ray tubes, mass spectrometers, and Hall effect sensors.

The Lorentz Force Law

The Lorentz force on a point charge q moving with velocity v in the presence of an electric field E and magnetic flux density B is given by F = q(E + v x B). The first term qE is the electric force, which acts along the direction of E for positive charges and opposite to E for negative charges. The second term q(v x B) is the magnetic force, which is always perpendicular to both the velocity v and the field B.

A critical distinction between these two components is that the electric force can do work on the charge (it can increase or decrease the kinetic energy), whereas the magnetic force does no work on the charge because it is always perpendicular to the velocity. The magnetic force can change the direction of motion but not the speed of the particle. This is why a charged particle in a pure magnetic field moves in a circular or helical path without gaining energy.

Circular Motion in a Magnetic Field

When a charged particle with charge q, mass m, and velocity v enters a region containing only a uniform magnetic field B perpendicular to v, it follows a circular path. The magnetic force provides the centripetal acceleration: qvB = mv^2 / r. Solving for the cyclotron radius (also called the Larmor radius) gives r = mv / (qB). The cyclotron frequency (angular frequency) is omega_c = qB / m, which is independent of velocity and is widely used in plasma physics and charged particle accelerators.

If the velocity has a component parallel to B in addition to a perpendicular component, the particle follows a helical path. The parallel component is unaffected by B (no force), while the perpendicular component causes circular motion. This helical motion governs the trapping of charged particles in the Van Allen belts of the Earth's magnetosphere.

Practical Understanding: Hall Effect and Velocity Selector

In the velocity selector, crossed electric and magnetic fields are arranged so that qE and q(v x B) exactly cancel for a specific velocity v = E/B. Particles with this velocity pass through undeflected while others are deflected. This principle is used in mass spectrometers to select ions of a specific velocity before separating them by mass.

The Hall effect arises directly from the Lorentz force: when current-carrying material is placed in a perpendicular magnetic field, the magnetic force on the charge carriers separates positive and negative charges to opposite faces, creating a transverse Hall voltage. The sign of the Hall voltage reveals whether the majority carriers are electrons or holes, making it a powerful tool for semiconductor characterization.

Given:
An electron (q = 1.6e-19 C, m = 9.11e-31 kg) moves with velocity v = 2e6 m/s
perpendicular to a uniform magnetic field B = 0.05 T.

Why this formula applies:
Only B is present, so F = qvB provides centripetal force mv2/r.
At equilibrium: qvB = mv2/r  -->  r = mv/(qB)

Formula:
r = mv / (qB)
omega_c = qB / m

Substitution:
r = (9.11e-31 * 2e6) / (1.6e-19 * 0.05)
omega_c = (1.6e-19 * 0.05) / 9.11e-31

Calculation:
Numerator: 9.11e-31 * 2e6 = 1.822e-24
Denominator: 1.6e-19 * 0.05 = 8e-21
r = 1.822e-24 / 8e-21 = 0.2278e-3 m = 0.228 mm
omega_c = 8e-21 / 9.11e-31 = 8.78e9 rad/s

Final Answer:
Cyclotron radius r = 0.228 mm
Cyclotron frequency omega_c = 8.78 x 10^9 rad/s

Exam Tip: The magnetic force q(v x B) does NO work on the particle. It changes direction but not speed. For circular motion in B alone: r = mv/(qB) and omega_c = qB/m. omega_c is independent of velocity, a GATE favourite fact.

• Lorentz force: F = q(E + v x B); electric part does work, magnetic part does not.
• In pure B field, particle follows circular path with r = mv/(qB) and cyclotron frequency omega_c = qB/m (independent of speed).
• Velocity component parallel to B is unaffected; combination of circular and linear motion gives helical path.
• Velocity selector: v = E/B when electric and magnetic forces balance.
• Hall effect: transverse voltage developed due to Lorentz force separating charge carriers; sign reveals carrier type.

Quick Revision

• F = q(E + v x B) is the complete Lorentz force expression.
• Magnetic force is always perpendicular to v; does no work, changes direction only.
• Cyclotron radius: r = mv/(qB); cyclotron frequency: omega_c = qB/m.
• omega_c is independent of velocity (key GATE fact).
• Velocity selector: v = E/B at equilibrium of electric and magnetic forces.
• Trap: The magnetic force does NO work; it cannot change kinetic energy, only trajectory direction.
• Helical motion occurs when v has both parallel and perpendicular components to B.