Lorentz force and its applications

Why it matters

The Lorentz force is the single law that tells a charge how to move in electric and magnetic fields. In instrumentation it is behind the Hall-effect sensor (current probes, position and speed sensors, gaussmeters), magnetic deflection in CRTs, mass spectrometers, velocity selectors and the force on the coil of a PMMC meter. Knowing how to apply it in vector form, and what it can and cannot do, avoids many sign and direction errors.

Key ideas

The law. A charge Q moving with velocity u in fields E and B feels F = Q(E + u × B). The electric part acts whether or not the charge moves and is along E (for positive Q). The magnetic part acts only on moving charges and is perpendicular to both u and B.

Work and energy. Because Q(u × B) is always perpendicular to u, the magnetic force does no work: it changes direction but never speed or kinetic energy. Only the electric field can change a particle's energy.

Motion in a uniform B.

• u perpendicular to B: uniform circular motion with radius r = mu/(|Q|B). The angular (cyclotron) frequency ω = |Q|B/m is independent of speed — the principle of the cyclotron.
• u at an angle to B: the parallel component is unaffected, the perpendicular component circulates, so the path is a helix.
• u parallel to B: no magnetic force; straight line.

Crossed fields. With E perpendicular to B and both perpendicular to u, the electric and magnetic forces cancel when u = E/B. Only particles at that speed pass undeflected — a velocity selector (Wien filter), used before a mass spectrometer.

Hall effect. A current I in a strip of thickness t across a perpendicular field B deflects the carriers sideways until the transverse electric field they build up balances the magnetic force (qE_H = quB). The Hall voltage across the strip is V_H = IB/(nqt), where n is the carrier density. Consequences:

• V_H is proportional to B (and to I), so the device measures B or, with a magnetic core around a conductor, current without contact.
• The sign of V_H reveals the sign of the carriers (n-type versus p-type).
• Semiconductors are used because their small n gives a much larger V_H than metals.

Force on currents. Summing Q u × B over the carriers in a wire gives dF = I dl × B, the force used in motors and meters (see the topic on magnetic force and torque).

Limits. These equations are non-relativistic (u ≪ c). Gravity is negligible for electrons and ions compared with typical electric and magnetic forces.

Formulas

• F = Q(E + u × B) — Lorentz force (N); Q in C, E in V/m, u in m/s, B in T.
• |F| = |Q| u B sin θ — magnetic part; θ between u and B.
• r = m u⊥ / (|Q| B) — radius of circular motion (m); m in kg.
• ω = |Q|B / m, f = |Q|B / (2πm), T = 2πm / (|Q|B) — cyclotron frequency and period.
• p = u∥ T = 2πm u∥ / (|Q|B) — pitch of a helix.
• u = E / B — velocity selector (crossed fields).
• V_H = I B / (n q t), R_H = 1 / (nq) — Hall voltage (V) and Hall coefficient (m³/C); n in m⁻³, t = thickness along B (m).
• Constants: e = 1.602 × 10⁻¹⁹ C; m_e = 9.109 × 10⁻³¹ kg; m_p = 1.673 × 10⁻²⁷ kg.

Worked examples

Example 1 (standard). A proton moves at 2 × 10⁶ m/s perpendicular to a uniform 0.5 T field. Find the force, the radius of its path and its cyclotron frequency.

1. F = quB = 1.6 × 10⁻¹⁹ × 2 × 10⁶ × 0.5 = 1.6 × 10⁻¹³ N.
2. r = mu/(qB) = 1.673 × 10⁻²⁷ × 2 × 10⁶ / (1.602 × 10⁻¹⁹ × 0.5) = 41.8 mm.
3. f = qB/(2πm) = 1.602 × 10⁻¹⁹ × 0.5 / (2π × 1.673 × 10⁻²⁷) = 7.62 MHz, independent of the speed.

Example 2 (GATE level, vector form). An electron (Q = −1.6 × 10⁻¹⁹ C) moves with u = 2 × 10⁶ a_x m/s through E = 10⁵ a_y V/m and B = 0.1 a_z T. Find the total force.

1. u × B = 2 × 10⁶ × 0.1 (a_x × a_z) = 2 × 10⁵ (−a_y) V/m, since a_x × a_z = −a_y.
2. E + u × B = (10⁵ − 2 × 10⁵) a_y = −10⁵ a_y V/m.
3. F = Q(E + u × B) = (−1.6 × 10⁻¹⁹)(−10⁵ a_y) = 1.6 × 10⁻¹⁴ a_y N.
4. Interpretation: the magnetic term is twice the electric term, so the electron is not undeflected; it would pass straight only at u = E/B = 10⁵/0.1 = 10⁶ m/s.

Example 3 (Hall sensor). A semiconductor Hall plate has thickness t = 0.5 mm and carrier density n = 10²¹ m⁻³, and carries I = 10 mA in a perpendicular B = 0.5 T. Find V_H and R_H, and compare with a copper strip of the same size (n = 8.5 × 10²⁸ m⁻³).

1. V_H = IB/(nqt) = 0.01 × 0.5 / (10²¹ × 1.6 × 10⁻¹⁹ × 5 × 10⁻⁴) = 0.005 / 0.08 = 62.5 mV.
2. R_H = 1/(nq) = 1 / (10²¹ × 1.6 × 10⁻¹⁹) = 6.25 × 10⁻³ m³/C.
3. Copper: V_H = 0.005 / (8.5 × 10²⁸ × 1.6 × 10⁻¹⁹ × 5 × 10⁻⁴) = 0.74 nV — about 10⁸ times smaller, which is why Hall sensors use semiconductors.

Common mistakes

• Forgetting that an electron's force is opposite to u × B.
• Getting cross-product orientation wrong: a_x × a_y = a_z, a_y × a_z = a_x, a_z × a_x = a_y; reversing the order flips the sign.
• Saying a magnetic field can accelerate a particle to higher speed (only E can).
• Using total speed instead of the perpendicular component in r = mu⊥/(qB).
• Taking t in the Hall formula as the width across which V_H is measured; it is the thickness along B.
• Mixing up the cyclotron frequency in rad/s (qB/m) and in Hz (qB/2πm).

For GATE IN

Expect: force on a charge from given E, B and u vectors; radius, period or pitch of motion in B; crossed-field velocity selector; Hall voltage, Hall coefficient, carrier density and mobility from Hall measurements (very common in instrumentation papers); and identifying the carrier type from the sign of V_H. Practise vector cross products until they are automatic.

Quick check

1. What is the magnetic force on a charge moving parallel to B?
2. If B is doubled, what happens to the radius of a charged particle's circular path?
3. Does the cyclotron frequency depend on the particle's speed?
4. E = 2 × 10⁴ V/m and B = 0.01 T are crossed. What speed passes undeflected?
5. Which materials give larger Hall voltages, metals or semiconductors?

Answers: 1. Zero. 2. It halves. 3. No (non-relativistic). 4. 2 × 10⁶ m/s. 5. Semiconductors (lower carrier density).