What is the Lorentz force on a moving charge?

When a particle with charge q moves at speed v through a magnetic field B at angle θ to the field, it experiences a force F = |q| · v · B · sin(θ). The direction is given by the right-hand rule (or the left-hand rule for negative charges). Maximum force occurs at θ = 90° and zero force when the particle moves parallel to the field (θ = 0°).

What is the force on a current-carrying wire in a magnetic field?

A wire of length L carrying current I in a magnetic field B experiences a force F = B · I · L · sin(θ), where θ is the angle between the current direction and B. This is the principle behind electric motors: current-carrying conductors in a magnet gap are pushed sideways, creating torque.

How is the force between two parallel wires calculated?

Two long parallel wires separated by distance d, carrying currents I₁ and I₂, exert a force per unit length F/L = μ₀ · I₁ · I₂ / (2π · d) on each other, where μ₀ = 4π×10⁻⁷ T·m/A. Currents in the same direction attract; opposite directions repel. This relationship historically defined the ampere (SI unit of current) before the 2019 SI revision.

What is μ₀ and what is its value?

μ₀ is the magnetic permeability of free space (vacuum). Its exact value is 4π×10⁻⁷ T·m/A ≈ 1.25664×10⁻⁶ T·m/A. In a magnetic material, replace μ₀ with μ₀ · μᵣ where μᵣ is the relative permeability of the medium (e.g. iron has μᵣ ≈ 200–5000).

Why is there no force when a charge moves parallel to B?

The force depends on the cross product of velocity and field: F = q(v × B). When v is parallel to B the cross product is zero, giving F = 0. This is why charged particles spiral along magnetic field lines rather than being deflected across them — the component of velocity along B is unchanged while the perpendicular component is continuously redirected in a circle.

Does this calculator give the direction of the force?

This calculator gives the magnitude only. Direction requires the right-hand rule (or left-hand rule for negative charges): point your fingers along v (or I), curl them toward B — your thumb points in the direction of force on a positive charge. The calculator notes whether the wire interaction is attractive or repulsive based on the current directions you select.

Magnetic Force Calculator

Magnetic forces govern electric motors, MRI scanners, mass spectrometers, particle accelerators, and the aurora borealis — wherever a moving charge or current-carrying conductor sits inside a magnetic field, a sideways force appears. This calculator handles three of the most important magnetic-force relationships in one tool: the Lorentz force on a single moving charge, the Ampere force on a current-carrying wire, and the force per unit length between two parallel conductors. Every mode lets you solve for any variable, shows the formula used, and displays a one-line worked calculation in SI units.

The three formulas

1. Lorentz force on a moving charge

When a particle with charge q moves at speed v through a uniform magnetic field B, and the angle between v and B is θ, the magnetic part of the Lorentz force has magnitude:

F = |q| · v · B · sin(θ)

where F is in newtons (N), q in coulombs (C), v in m/s, B in tesla (T), and θ in degrees. The force is zero when the particle travels parallel to the field, and maximum when it travels perpendicular (θ = 90°). Direction follows the right-hand rule: fingers along v, curl toward B, thumb points toward the force on a positive charge.

Rearrangements: q = F / (v · B · sin θ), v = F / (|q| · B · sin θ), B = F / (|q| · v · sin θ), θ = arcsin(F / (|q| · v · B)).

2. Force on a current-carrying wire

A straight wire of length L carrying current I at angle θ to a uniform field B experiences:

F = B · I · L · sin(θ)

This is the working principle of every DC electric motor. When θ = 90° the force is maximised; at θ = 0° the wire is parallel to B and no force acts. Rearranging gives I = F / (B · L · sin θ) and L = F / (B · I · sin θ).

3. Force per unit length between two parallel wires

Two infinitely long parallel wires carrying currents I₁ and I₂ separated by distance d interact via their own magnetic fields. The force per unit length between them is:

F/L = μ₀ · I₁ · I₂ / (2π · d)

where μ₀ = 4π×10⁻⁷ T·m/A is the permeability of free space. Currents in the same direction produce an attractive force; opposite-direction currents repel. This formula was historically used to define the ampere — 1 A was the current that produces exactly 2×10⁻⁷ N/m between wires 1 m apart.

Worked examples

Lorentz on a charge: An alpha particle (q = +3.204×10⁻¹⁹ C) moves at 2×10⁷ m/s perpendicular to a 0.3 T field (θ = 90°). Select “Lorentz force on a moving charge”, set q = 3.204×10⁻¹⁹ C (or 0.3204 nC), v = 2×10⁷ m/s, B = 0.3 T, θ = 90°. Result: F = 1.922×10⁻¹² N — a piconewton-scale deflection that curves the particle into a circular arc inside a detector.

Wire in a motor: A wire 0.15 m long carries 8 A at 90° to a 0.25 T field. Select “Force on a current-carrying wire”, enter I = 8 A, L = 0.15 m, B = 0.25 T, θ = 90°. Result: F = 0.3 N — the force that pushes the rotor windings and generates shaft torque.

Parallel bus bars: Two bus bars 5 cm apart each carry 500 A in the same direction. Select “Force between two parallel wires”, enter I₁ = I₂ = 500 A, d = 5 cm, same direction. Result: F/L ≈ 1 N/m of attractive force — an important structural load when designing switchgear.

Physical constants used

Constant	Symbol	Value
Permeability of free space	μ₀	4π×10⁻⁷ T·m/A
Elementary charge	e	1.60217663×10⁻¹⁹ C

All constants follow CODATA 2018 recommended values. The calculator uses magnitudes throughout; apply the right-hand rule separately to determine direction.