CalcMagnetic Force on Charge Calculator
Calculate the magnetic force on a moving charge.
What is Magnetic Force on a Charge?
When an electric charge moves through a magnetic field, it experiences a force perpendicular to both its velocity and the magnetic field direction. This magnetic force, described by the Lorentz force law, governs the motion of electrons in TV tubes, protons in particle accelerators, and ions in mass spectrometers. Unlike electric force, magnetic force acts only on moving charges and never does work — it changes direction but not speed.
Picture an electron (q = -1.6×10⁻¹⁹ C) traveling at 1 million meters per second through a magnetic field of 0.5 tesla (a strong lab magnet). If the electron moves perpendicular to the field, the force is F = |q|vB = (1.6×10⁻¹⁹) × (10⁶) × (0.5) = 8×10⁻¹⁴ N. This tiny force causes the electron to curve in a circular path with radius r = mv/(|q|B) = (9.11×10⁻³¹ × 10⁶)/(1.6×10⁻¹⁹ × 0.5) = 1.14×10⁻⁵ m — about 11 micrometers. This is how cathode ray tubes steer electron beams to paint images on screens.
The magnetic force explains why charged particles spiral along Earth's magnetic field lines, creating the aurora borealis. Solar wind protons and electrons get trapped in the magnetosphere, spiraling around field lines and colliding with atmospheric gases near the poles. The same principle confines plasma in fusion reactors and guides particles through the Large Hadron Collider's 27-kilometer ring using superconducting magnets producing 8+ tesla fields.
How it Works: Formulas Explained
The magnetic force on a moving charge is F = qvB sin(θ), where q is the charge in coulombs, v is the speed in m/s, B is the magnetic field strength in tesla, and θ is the angle between the velocity vector and the magnetic field vector. Maximum force occurs at θ = 90° (perpendicular motion): F_max = |q|vB. Zero force occurs at θ = 0° or 180° (parallel motion) — charges moving along field lines feel no magnetic force.
The force direction follows the right-hand rule for positive charges: point fingers in the velocity direction, curl them toward the magnetic field direction, and your thumb points in the force direction. For negative charges (electrons, negative ions), the force points opposite to the right-hand rule prediction. An alternative: point thumb along v, fingers along B, and palm pushes in the force direction for positive charges.
Because the force is always perpendicular to velocity, it causes circular or helical motion without changing speed. The centripetal force mv²/r equals the magnetic force |q|vB, giving the gyroradius (cyclotron radius): r = mv/(|q|B). The angular frequency ω = v/r = |q|B/m is the cyclotron frequency — independent of velocity! This property enables cyclotrons to accelerate particles with a fixed-frequency RF field.
Working through a complete example: A proton (q = +1.6×10⁻¹⁹ C, m = 1.67×10⁻²⁷ kg) enters a 2.0 T magnetic field at v = 5×10⁶ m/s, perpendicular to the field. Force: F = (1.6×10⁻¹⁹) × (5×10⁶) × (2.0) = 1.6×10⁻¹² N. Radius: r = mv/(qB) = (1.67×10⁻²⁷ × 5×10⁶)/(1.6×10⁻¹⁹ × 2.0) = 8.35×10⁻²¹/3.2×10⁻¹⁹ = 0.026 m = 2.6 cm. The proton spirals in a 2.6 cm radius circle, completing one revolution in T = 2πm/(qB) = 3.28×10⁻⁸ s — about 30 MHz cyclotron frequency.
Step-by-Step Guide
- Identify the charge, velocity, and magnetic field.** Determine q (with sign), v (magnitude and direction), and B (magnitude and direction). Common charges: proton q = +1.6×10⁻¹⁹ C, electron q = -1.6×10⁻¹⁹ C, alpha particle q = +3.2×10⁻¹⁹ C. Velocities in particle physics range from 10⁵ m/s (thermal ions) to nearly c = 3×10⁸ m/s (relativistic particles).
- Determine the angle θ between v and B.** If velocity and field are perpendicular, θ = 90° and sin(θ) = 1, giving maximum force. If parallel or antiparallel, θ = 0° or 180° and sin(θ) = 0, giving zero force. For intermediate angles, use the actual angle. A particle moving at 45° to the field experiences F = |q|vB sin(45°) = 0.707|q|vB — about 71% of maximum.
- Calculate the force magnitude.** F = |q|vB sin(θ). Example: An alpha particle (q = +3.2×10⁻¹⁹ C) travels at v = 2×10⁷ m/s through B = 1.5 T at θ = 60°. F = (3.2×10⁻¹⁹) × (2×10⁷) × (1.5) × sin(60°) = 9.6×10⁻¹² × 0.866 = 8.31×10⁻¹² N. Use absolute value of charge for magnitude; sign determines direction.
- Find the force direction using the right-hand rule.** For positive charges: right hand, fingers along v, curl toward B, thumb gives F direction. For negative charges: either use left hand with same rule, or use right hand and reverse the result. If v points east and B points north, force on a positive charge points upward; on an electron, downward.
- Calculate the trajectory radius if needed.** For perpendicular motion, r = mv/(|q|B). A 1 keV electron (v ≈ 1.87×10⁷ m/s) in B = 0.01 T has r = (9.11×10⁻³¹ × 1.87×10⁷)/(1.6×10⁻¹⁹ × 0.01) = 1.07×10⁻² m ≈ 1 cm. Higher energy or weaker field gives larger radius. This radius determines the size of cyclotrons and spectrometers.
- Account for combined electric and magnetic fields.** When both fields are present, total force is the Lorentz force: F = qE + qv×B. If E and v×B are opposite and equal in magnitude, they cancel — this is the velocity selector principle. Particles with v = E/B pass through undeflected; others curve away. Mass spectrometers use this to filter particles by velocity before magnetic analysis.
Real-World Examples
Example 1: Mass spectrometer.** A mass spectrometer ionizes molecules and accelerates them through a known voltage, giving all ions the same kinetic energy. They then enter a magnetic field that bends them into circular paths. Radius r = mv/(qB). Since all ions have the same energy (½mv² = qV), velocity v = √(2qV/m). Substituting: r = √(2mV/q)/B. Heavier ions curve less (larger radius). A protein ion with m/z = 1000 in B = 1 T after 5000 V acceleration: r = √(2×1000×1.66×10⁻²⁷×5000/(1.6×10⁻¹⁹))/1 = 0.18 m. Measuring r reveals the mass-to-charge ratio.
Example 2: Cyclotron particle accelerator.** A cyclotron uses a constant magnetic field and oscillating electric field to accelerate protons. With B = 1.5 T, the cyclotron frequency is f = qB/(2πm) = (1.6×10⁻¹⁹ × 1.5)/(2π × 1.67×10⁻²⁷) = 22.9 MHz. The RF field oscillates at this frequency, giving protons a kick each half-revolution. At radius 0.5 m, proton speed is v = qBr/m = (1.6×10⁻¹⁹ × 1.5 × 0.5)/(1.67×10⁻²⁷) = 7.19×10⁷ m/s — about 24% light speed, kinetic energy 27 MeV.
Example 3: Aurora borealis formation.** Solar wind protons (v ≈ 4×10⁵ m/s) encounter Earth's magnetic field (B ≈ 5×10⁻⁵ T at the surface). The gyroradius is r = mv/(qB) = (1.67×10⁻²⁷ × 4×10⁵)/(1.6×10⁻¹⁹ × 5×10⁻⁵) = 83,500 m — about 84 km. Particles spiral along field lines toward the poles, where the field converges. Collisions with oxygen (green light at 557.7 nm) and nitrogen (purple/red) create the auroral display at altitudes of 100-400 km.
Example 4: Hall effect sensor.** When current flows through a conductor in a magnetic field, charge carriers experience magnetic force sideways, creating a voltage across the conductor (Hall voltage). For electrons drifting at v_d = 10⁻⁴ m/s in a 0.1 T field: F = qv_dB = 1.6×10⁻¹⁹ × 10⁻⁴ × 0.1 = 1.6×10⁻²⁴ N per electron. This tiny force, summed over many carriers, creates a measurable voltage proportional to B. Hall sensors use this principle for contactless position sensing and current measurement.
Example 5: Magnetron in microwave ovens.** A magnetron generates microwaves using electrons spiraling in a magnetic field. Electrons emitted from a central cathode accelerate toward an anode while a strong magnetic field (B ≈ 0.1 T) forces them into circular paths. At radius r = 2 cm, electron speed is v = qBr/m = (1.6×10⁻¹⁹ × 0.1 × 0.02)/(9.11×10⁻³¹) = 3.51×10⁸ m/s — relativistic! The actual speed is limited to below c, but the cyclotron motion causes electrons to emit 2.45 GHz microwaves, which heat food by exciting water molecules.
Common Mistakes to Avoid
Forgetting that magnetic force requires motion.** A stationary charge in a magnetic field experiences zero force. The v in F = qvB sin(θ) must be nonzero. If a problem states "a charge is placed in a magnetic field" without mentioning motion, the magnetic force is zero. Electric fields exert force on stationary charges; magnetic fields do not. This is a fundamental distinction between electric and magnetic interactions.
Misapplying the right-hand rule.** The right-hand rule gives force direction for positive charges only. For electrons or negative ions, either use your left hand with the same finger assignments, or use your right hand and flip the result. A common error: getting the correct magnitude but wrong direction because the charge sign was ignored. Label charges clearly and double-check direction for negative charges.
Confusing magnetic force with magnetic torque.** Force F = qvB sin(θ) acts on moving charges. Torque τ = μB sin(θ) acts on magnetic dipoles (current loops, bar magnets). They're different phenomena. A current-carrying wire in a magnetic field experiences force F = ILB sin(θ), where I is current and L is wire length — this is the sum of forces on individual moving charges in the wire.
Assuming magnetic force changes particle speed.** Magnetic force is always perpendicular to velocity, so it does zero work (W = F·d = 0 when F ⊥ d). Kinetic energy and speed remain constant; only direction changes. If a particle speeds up or slows down in a "magnetic" field, either an electric field is also present, or the magnetic field is changing with time (inducing an electric field via Faraday's law).
Pro Tips
Use the velocity selector principle.** When electric and magnetic fields are crossed (perpendicular to each other) and a particle passes through undeflected, the forces balance: qE = qvB, giving v = E/B. This velocity selector filters particles by speed regardless of mass or charge. All particles with v = E/B pass straight through; faster particles curve one way, slower particles curve the other. Essential first stage in mass spectrometers.
Recognize helical motion from angled entry.** When a charge enters a magnetic field at an angle θ (not 90°), decompose velocity into perpendicular (v⊥ = v sin(θ)) and parallel (v∥ = v cos(θ)) components. The perpendicular component causes circular motion; the parallel component continues unchanged. Result: helical (spiral) motion along field lines. Pitch of the helix (distance per turn) is p = v∥ × T = v cos(θ) × 2πm/(qB).
Apply the cyclotron frequency shortcut.** For circular motion in a magnetic field, angular frequency ω = |q|B/m depends only on charge-to-mass ratio and field strength — not on velocity or radius! This counterintuitive result (faster particles move in larger circles but take the same time per revolution) enables cyclotrons to use fixed-frequency RF acceleration. Frequency f = ω/(2π) = |q|B/(2πm). For protons in 1 T: f ≈ 15.3 MHz.
Estimate using order-of-magnitude values.** Electron mass ~10⁻³⁰ kg, proton mass ~10⁻²⁷ kg, elementary charge ~10⁻¹⁹ C, lab magnetic fields ~0.1-10 T. A 1 MeV proton (v ~10⁷ m/s) in 1 T field: r = mv/qB ~ 10⁻²⁷ × 10⁷ / (10⁻¹⁹ × 1) = 0.1 m. An electron under same conditions: r ~ 10⁻³⁰ × 10⁷ / (10⁻¹⁹ × 1) = 10⁻⁴ m — much tighter spiral due to lower mass. Quick estimates catch calculation errors.
FAQs
Work equals force times displacement in the force direction: W = F·d. Magnetic force is always perpendicular to velocity (and thus to the instantaneous displacement). The dot product of perpendicular vectors is zero, so W = 0. Magnetic forces can redirect particles but cannot speed them up or slow them down. To change a charged particle's energy, you need an electric field or a time-varying magnetic field (which induces an electric field).
B (magnetic flux density, measured in tesla) is the fundamental magnetic field that appears in the Lorentz force. H (magnetic field strength, measured in A/m) is an auxiliary field useful in materials: B = μH where μ is permeability. In vacuum, B = μ₀H with μ₀ = 4π×10⁻⁷ T·m/A. For force calculations, always use B. H is mainly used in magnetostatics problems involving magnetic materials.
Earth's magnetic field is about 50 microtesla (0.00005 T). A refrigerator magnet is about 0.005 T (5 mT). A strong lab electromagnet produces 1-2 T. MRI machines use 1.5-3 T superconducting magnets. The strongest continuous lab fields reach about 45 T. Pulsed magnets can briefly exceed 100 T. Neutron stars have fields of 10⁸ to 10¹¹ T — strong enough to distort atomic orbitals and make chemistry impossible.
Yes — magnetic bottles and Earth's magnetosphere trap charged particles. Particles spiral along field lines and reflect where the field strengthens (magnetic mirror effect). Earth's Van Allen belts trap solar wind particles between mirror points in the northern and southern hemispheres. Fusion reactors like tokamaks use toroidal (doughnut-shaped) magnetic fields to confine hot plasma, preventing it from touching the reactor walls.
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