Lorentz Force — Compute magnetic force on a moving charge (Lorentz force) - Physics 2

SECTION 1

Historical Context & Motivation

The relationship between electricity and magnetism puzzled natural philosophers for centuries before a unified mathematical framework emerged. Early observations that lodestones could deflect compass needles hinted at deep connections between moving charges and magnetic phenomena, but it was not until the nineteenth century that experimentalists could probe these connections quantitatively. The Lorentz force law, which describes the force exerted on a charged particle moving through electric and magnetic fields, represents the culmination of decades of experimental and theoretical work by some of the most celebrated figures in physics.

1820 • Ørsted's Discovery

Hans Christian Ørsted demonstrates that an electric current deflects a nearby compass needle, establishing the first concrete link between electricity and magnetism and igniting a wave of research across Europe.

1831 • Faraday's Induction

Michael Faraday discovers electromagnetic induction—showing that a changing magnetic field can produce an electric current—and introduces the concept of 'lines of force,' a precursor to the modern field concept.

1865 • Maxwell's Equations

James Clerk Maxwell publishes his unified field equations, mathematically encoding the interplay of electric and magnetic fields and predicting electromagnetic waves.

1892 • Lorentz Formulates the Force Law

Hendrik Antoon Lorentz synthesizes earlier results into a single expression for the total electromagnetic force on a point charge, F = q(E + v × B), providing the foundation for modern electrodynamics.

1897 • Thomson Measures e/m

J. J. Thomson uses crossed electric and magnetic fields—direct applications of the Lorentz force—to measure the charge-to-mass ratio of the electron, confirming the existence of subatomic particles.

With Lorentz's formulation in hand, physicists could finally answer a deceptively simple question: how does a magnetic field push a charged particle that moves through it? The magnetic component of the Lorentz force is fundamentally different from familiar contact forces or even the Coulomb electric force because it depends on the particle's velocity and acts perpendicular to both the velocity and the field. Understanding this cross-product structure is the central goal of this lesson.

SECTION 2

Core Principles & Definitions

Before diving into calculations, it is essential to internalize the foundational ideas that govern the magnetic force on a moving charge. These principles determine when the force exists, in what direction it acts, and what it can and cannot do to a charged particle's energy. Each principle below carries physical consequences that you will encounter repeatedly in electromagnetism courses and in practical applications ranging from mass spectrometry to magnetic confinement fusion.

Velocity Dependence

The magnetic force on a charge is proportional to the charge's velocity. A stationary charge in a purely magnetic field experiences no magnetic force whatsoever—only moving charges feel the push.

Cross-Product Direction

The force is given by the cross product v × B, which means it is always perpendicular to both the velocity vector and the magnetic field vector. This geometric relationship is captured by the right-hand rule.

No Work Done

Because F is perpendicular to v, the magnetic force does zero work on the particle: W = F · ds = 0. It can change the direction of motion but never the speed or kinetic energy.

Charge-Sign Sensitivity

The force direction reverses for negative charges. A proton and an electron moving in the same direction through the same field are deflected in opposite directions, which is the operating principle of velocity selectors and mass spectrometers.

Angle Dependence

The magnitude scales as sin θ, where θ is the angle between v and B. Maximum force occurs when v ⊥ B (θ = 90°), and zero force occurs when v ∥ B (θ = 0° or 180°).

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KEY TAKEAWAY

Think of the magnetic force as a cosmic referee that can redirect a charged particle's path but never speed it up or slow it down. Just as a banked curve on a racetrack changes a car's direction without altering its speedometer reading, the magnetic force acts as a centripetal-like deflector—always perpendicular to the motion, always steering, never doing work.

SECTION 3

Visual Explanation — The Right-Hand Rule & Force Geometry

The geometry of the magnetic Lorentz force is intrinsically three-dimensional: you need three mutually relevant vectors (velocity, magnetic field, and force) to describe the situation fully. The diagram below illustrates the canonical scenario of a positive charge moving to the right through a magnetic field directed into the page. The resulting force, determined by the right-hand rule, points upward—perpendicular to both v and B.

A positive charge (pink circle) moves to the right with velocity v (cyan arrow) through a uniform magnetic field B directed into the page (× symbols). The resulting Lorentz force F (green arrow) points upward, perpendicular to both v and B. For a negative charge, the force would reverse to point downward.

Notice that the three vectors—v, B, and F—form a right-handed triad for positive charges. If the charge were negative, you would either use the left-hand rule or apply the right-hand rule and then reverse the resulting direction. In the diagram, the × symbols indicate magnetic field lines pointing into the plane of the screen (the tail of an arrow); dots (·) would indicate field lines coming out of the screen (the tip of an arrow). This convention is ubiquitous in electromagnetism problems and is worth committing to memory.

SECTION 4

Mathematical Framework

The full Lorentz force law encompasses both the electric and magnetic forces acting on a point charge. In many magnetism problems the electric field is absent or treated separately, so we focus on the magnetic term. The vector and scalar forms of the magnetic force, together with the SI unit analysis, form the core mathematical toolkit for this topic.

FULL LORENTZ FORCE

F = q(E + v × B)

F = total electromagnetic force (N), q = charge of the particle (C), E = electric field (N/C), v = velocity of the charge (m/s), B = magnetic field (T). When E = 0, the force is purely magnetic.

MAGNETIC FORCE (VECTOR FORM)

F_B = qv × B

The cross product ensures that F_B is perpendicular to the plane defined by v and B. Direction is given by the right-hand rule for positive q and reversed for negative q.

MAGNETIC FORCE (MAGNITUDE)

|F_B| = |q| v B sin θ

Here θ is the angle between v and B. Maximum force at θ = 90°; zero force at θ = 0° or 180°. The absolute value of q ensures magnitude is always non-negative.

CIRCULAR MOTION RADIUS

r = mv / (|q|B)

When v ⊥ B, the magnetic force provides centripetal acceleration, and the particle moves in a circle of radius r. Here m = particle mass (kg). This result is derived by setting |q|vB = mv²/r and solving for r.

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Unit Check

The SI unit of magnetic field B is the tesla (T). Verify: [C] × [m/s] × [T] = [C] × [m/s] × [kg/(A·s²)] = [kg·m/s²] = [N]. This dimensional check confirms the force formula is consistent.

SECTION 5

Charged-Particle Trajectories in Magnetic Fields

Because the magnetic force is always perpendicular to the velocity, it naturally produces curved trajectories. The specific shape of the path depends on the angle between the initial velocity vector and the magnetic field. Three canonical cases arise, each with important practical significance.

Three canonical trajectories: Case 1 — when v is perpendicular to B the charge traces a circle. Case 2 — when v is parallel to B the force vanishes and the charge moves in a straight line. Case 3 — when v has components both parallel and perpendicular to B the charge follows a helix, combining circular motion in the transverse plane with uniform drift along the field direction.

In the helical case, you can decompose the velocity into components parallel (v_∥ = v cos θ) and perpendicular (v_⊥ = v sin θ) to B. The perpendicular component governs the circular radius r = mv_⊥/(|q|B), while the parallel component carries the particle along the field line at constant speed. The combined motion produces a helix whose pitch (distance advanced per revolution) equals v_∥ × T, where T = 2πm/(|q|B) is the cyclotron period. Helical motion is responsible for charged particles spiraling along Earth's magnetic field lines and producing auroras at the poles.

Condition	θ	Trajectory	Force Magnitude
v ⊥ B	90°	Circle	\|q\|vB (maximum)
v ∥ B	0° or 180°	Straight line	0 (no force)
v at angle θ to B	0° < θ < 90°	Helix	\|q\|vB sin θ

Summary of trajectory types based on the angle between v and B

SECTION 6

Worked Example — Proton in a Uniform Magnetic Field

A proton (q = +1.60 × 10⁻¹⁹ C, m = 1.67 × 10⁻²⁷ kg) enters a region of uniform magnetic field B = 0.50 T directed in the +z-direction. The proton's velocity is v = (3.0 × 10⁵ m/s) x̂ + (4.0 × 10⁵ m/s) ẑ. Determine: (a) the magnetic force on the proton, (b) its magnitude, (c) the radius of the circular component of its helical path, and (d) the pitch of the helix.

Proton in a Helical Trajectory

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Step 1 — Identify Given Values

q = +1.60 × 10⁻¹⁹ C, m = 1.67 × 10⁻²⁷ kg, B = 0.50 T ẑ. The velocity components are v_x = 3.0 × 10⁵ m/s (perpendicular to B) and v_z = 4.0 × 10⁵ m/s (parallel to B). The speed is v = √(v_x² + v_z²) = 5.0 × 10⁵ m/s.

v_⊥ = 3.0 × 10⁵ m/s, v_∥ = 4.0 × 10⁵ m/s
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Step 2 — Compute the Cross Product v × B

v × B = (v_x x̂ + v_z ẑ) × (B ẑ). Using the determinant method: x̂ × ẑ = −ŷ, ẑ × ẑ = 0. Therefore v × B = v_x B (x̂ × ẑ) = v_x B (−ŷ) = −(3.0 × 10⁵)(0.50) ŷ = −1.5 × 10⁵ ŷ (T·m/s).

v × B = −1.5 × 10⁵ ŷ T·m/s
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Step 3 — Compute the Magnetic Force (part a)

F_B = qv × B = (1.60 × 10⁻¹⁹)(−1.5 × 10⁵) ŷ = −2.4 × 10⁻¹⁴ ŷ N. The force is in the −y-direction.

$F_B$ = −2.4 × 10⁻¹⁴ ŷ N
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Step 4 — Magnitude of the Force (part b)

|F_B| = |q| v_⊥ B = (1.60 × 10⁻¹⁹)(3.0 × 10⁵)(0.50) = 2.4 × 10⁻¹⁴ N. Alternatively, |q|vB sin θ where θ = arctan(v_x/v_z) ≈ 36.87° gives the same result since v sin θ = v_⊥.

| $F_B$ | = 2.4 × 10⁻¹⁴ N
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Step 5 — Radius of Circular Motion (part c)

r = mv_⊥ / (|q|B) = (1.67 × 10⁻²⁷)(3.0 × 10⁵) / [(1.60 × 10⁻¹⁹)(0.50)] = 5.01 × 10⁻²² / 8.0 × 10⁻²⁰ ≈ 6.26 × 10⁻³ m ≈ 6.3 mm.

r ≈ 6.3 mm
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Step 6 — Pitch of the Helix (part d)

First find the cyclotron period: T = 2πm / (|q|B) = 2π(1.67 × 10⁻²⁷) / [(1.60 × 10⁻¹⁹)(0.50)] = 1.049 × 10⁻²⁶ / 8.0 × 10⁻²⁰ ≈ 1.31 × 10⁻⁷ s. The pitch p = v_∥ × T = (4.0 × 10⁵)(1.31 × 10⁻⁷) ≈ 5.24 × 10⁻² m ≈ 5.2 cm.

Pitch ≈ 5.2 cm

SECTION 7

Applications, Strengths & Limitations

The magnetic Lorentz force underpins a remarkable range of technologies and natural phenomena. At the same time, it has well-defined limitations as a classical, non-relativistic expression. Understanding both its power and its boundaries is essential for applying it correctly and knowing when more advanced formulations are needed.

Strengths / Applications	Limitations / Caveats
Explains charged-particle deflection in mass spectrometers, enabling isotope identification	Applies only to point charges; for current-carrying wires, the force is F = IL × B (derived from Lorentz force but distinct in form)
Governs cyclotron and synchrotron operation for particle acceleration	At relativistic speeds, mass m must be replaced by γm (Lorentz factor), altering the radius formula
Describes the Hall effect, used to measure magnetic field strength and charge carrier concentration in semiconductors	Does not account for radiation emitted by accelerating charges (synchrotron radiation), which requires electrodynamic treatment
Explains auroral phenomena—charged solar-wind particles spiral along Earth's field lines and excite atmospheric gases	Assumes externally imposed fields; self-consistent field effects (plasma physics) require more complex models
Foundation for velocity selectors (crossed E and B fields) used in Thomson's e/m experiment	Neglects quantum-mechanical effects (spin-magnetic moment coupling, Landau levels) relevant at atomic scales

Practical strengths and theoretical limitations of the magnetic Lorentz force

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BROADER CONTEXT

The Lorentz force is the bridge between electrostatics and magnetism—it tells you precisely how fields affect charges in motion. In engineering, it drives electric motors (forces on current-carrying conductors), CRT displays (electron beam steering), and MRI machines (proton precession in strong fields). Every time a charged particle curves in a magnetic field, the Lorentz force is the mechanism at work.

SECTION 8

Connection to Relativistic & Quantum Electrodynamics

The classical Lorentz force law you have learned is the non-relativistic limit of a more general covariant expression in special relativity. When particle speeds approach the speed of light, the distinction between electric and magnetic fields blurs—what appears as a purely magnetic force in one reference frame may include an electric component in another. This observer-dependence is encoded in the electromagnetic field tensor F^μν, and the equation of motion becomes dp^μ/dτ = qF^μνu_ν. At the quantum level, particles interact with the electromagnetic field through virtual photon exchange, described by quantum electrodynamics (QED)—the most precisely tested theory in all of physics.

Feature	Classical Lorentz Force	Relativistic / QED Extension
Velocity regime	v ≪ c	Any v < c
Mass treatment	Constant rest mass m	Relativistic momentum p = γmv
E and B mixing	Treated as independent	Components of one field tensor F^μν; frame-dependent decomposition
Radiation	Ignored	Synchrotron radiation from Larmor formula; radiation reaction
Quantum effects	None	Landau levels, anomalous magnetic moment, virtual photon exchange

Classical vs. relativistic/quantum treatments of the electromagnetic force on a charge

For most problems in an introductory Physics 2 course—including mass spectrometers, cyclotrons, and the Hall effect—the classical Lorentz force is entirely sufficient. However, awareness of these extensions provides valuable context: the fact that magnetism itself is a relativistic effect (arising from length contraction of charge densities in moving frames) deepens your understanding of why the force depends on velocity in the first place.

SECTION 9

Practice Problems

Question 1: CONCEPTUAL

An electron moves due north through a region where the magnetic field points vertically upward. In what direction is the magnetic force on the electron? Explain why the answer differs from what you would obtain for a proton moving in the same direction.

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Apply the right-hand rule first as if the charge were positive: fingers point north (v), curl them upward toward B—the cross product v × B points west. However, the electron carries a negative charge, so the actual force is reversed: it points east. A proton moving in the same direction would feel a force to the west. This charge-sign sensitivity is precisely what allows mass spectrometers and velocity selectors to separate particles by charge polarity and mass.

Question 2: BASIC CALCULATION

A singly charged ion (q = +1.60 × 10⁻¹⁹ C) moves at 2.0 × 10⁶ m/s perpendicular to a uniform magnetic field of 0.80 T. Calculate the magnitude of the magnetic force on the ion.

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Since v ⊥ B, sin θ = 1 and the magnitude formula gives |F| = |q|vB sin 90° = (1.60 × 10⁻¹⁹)(2.0 × 10⁶)(0.80)(1) = 2.56 × 10⁻¹³ N. This force is tiny in absolute terms, but because the ion's mass is also extremely small (on the order of 10⁻²⁶ kg), the resulting acceleration is enormous—roughly 10¹³ m/s², sufficient to curve the ion's path into a tight circle inside a mass spectrometer.

Question 3: INTERMEDIATE

A proton (m = 1.67 × 10⁻²⁷ kg, q = +1.60 × 10⁻¹⁹ C) enters a uniform magnetic field B = 1.2 T with a velocity of 5.0 × 10⁶ m/s at an angle of 30° to the field. Determine (a) the radius of the helical path, (b) the cyclotron period, and (c) the pitch of the helix.

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(a) The perpendicular component is v⊥ = v sin 30° = (5.0 × 10⁶)(0.50) = 2.5 × 10⁶ m/s. Radius r = mv⊥/(|q|B) = (1.67 × 10⁻²⁷)(2.5 × 10⁶)/[(1.60 × 10⁻¹⁹)(1.2)] = 4.175 × 10⁻²¹ / 1.92 × 10⁻¹⁹ ≈ 2.17 × 10⁻² m ≈ 2.2 cm. (b) T = 2πm/(|q|B) = 2π(1.67 × 10⁻²⁷)/(1.92 × 10⁻¹⁹) ≈ 5.46 × 10⁻⁸ s. (c) v∥ = v cos 30° = (5.0 × 10⁶)(0.866) = 4.33 × 10⁶ m/s. Pitch = v∥T = (4.33 × 10⁶)(5.46 × 10⁻⁸) ≈ 0.237 m ≈ 23.7 cm. Note that the cyclotron period is independent of speed—a key fact exploited in cyclotron design.

Question 4: APPLIED

In a velocity selector, a beam of ions passes through a region where a uniform electric field E = 4.0 × 10⁴ N/C and a uniform magnetic field B = 0.20 T are perpendicular to each other and to the beam. Only ions with a specific speed pass through undeflected. (a) Derive an expression for the selected speed. (b) Calculate its value. (c) If the ions then enter a second region with only magnetic field B₂ = 0.35 T and trace a semicircle of diameter 16.4 cm, determine the ion's mass (given q = +1.60 × 10⁻¹⁹ C).

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(a) For an undeflected ion, the electric and magnetic forces balance: qE = qvB → v = E/B. This result is independent of q and m. (b) v = (4.0 × 10⁴)/(0.20) = 2.0 × 10⁵ m/s. (c) In the second region the semicircle diameter is 2r = 0.164 m, so r = 0.082 m. From r = mv/(|q|B₂), we get m = |q|B₂r/v = (1.60 × 10⁻¹⁹)(0.35)(0.082)/(2.0 × 10⁵) = 4.592 × 10⁻²¹ / 2.0 × 10⁵ ≈ 2.30 × 10⁻²⁶ kg. Dividing by the atomic mass unit (1.66 × 10⁻²⁷ kg), this is about 13.8 u, close to ¹⁴N, suggesting the ions are nitrogen.

Question 5: CRITICAL THINKING

A magnetic field cannot change a charged particle's kinetic energy because the magnetic force does no work. Yet a betatron—a type of particle accelerator—uses a changing magnetic field to accelerate electrons to high energies. How is this possible without violating the principle that the magnetic force does zero work? Discuss the role of Faraday's law in resolving this apparent paradox.

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The key distinction is between the magnetic force on a moving charge (F = qv × B) and the electric field induced by a time-varying magnetic field (Faraday's law: ∮ E · dl = −dΦ_B/dt). In a betatron, the magnetic field is increased over time, and by Faraday's law this produces a tangential induced electric field that acts along the circular orbit. It is this induced E-field—not the magnetic force—that does work on the electrons and increases their kinetic energy. The static magnetic component merely keeps the electrons on their circular path (providing centripetal force). Thus there is no violation: the magnetic force itself remains perpendicular to v and does zero work, while the induced electric field (a consequence of the changing B) is parallel to v and accelerates the particles. This problem illustrates the deep interplay between electric and magnetic phenomena described by Maxwell's equations.

SUMMARY

Lesson Summary

Summary

The Lorentz force law, F = q(E + v × B), encapsulates the total electromagnetic force on a charged particle. The magnetic component F_B = qv × B is unique in that it depends on the particle's velocity, acts perpendicular to both v and B, and therefore performs zero work on the charge. Its magnitude is |q|vB sin θ, reaching a maximum when v ⊥ B and vanishing when v ∥ B. The direction is given by the right-hand rule for positive charges (reversed for negative charges).

When a charged particle moves perpendicular to a uniform field, it follows a circular orbit with radius r = mv/(|q|B); when velocity has both perpendicular and parallel components, the trajectory is a helix. These principles underpin mass spectrometers, cyclotrons, velocity selectors, and the Hall effect. At relativistic speeds or quantum scales, the classical formula gives way to the covariant formulation or QED, but for the vast majority of introductory problems, F = qv × B is both elegant and exact.