AkCalculators

Magnetic force and the Lorentz law — moving charges, current-carrying conductors, vectors and solenoids

The magnetic force is central to electromagnetism and underpins technologies from electric motors to particle accelerators. The magnetic part of the Lorentz force gives the force experienced by a moving electric charge in a magnetic field. For extended current-carrying conductors similar formulas apply that are directly derived from the Lorentz law. This page collects practical formulas, vector methods, the commonly used solenoid field approximation and examples to help you compute forces in laboratory and engineering contexts.

The Lorentz force on a moving charge

For a particle with electric charge q moving with velocity v in a magnetic field B, the magnetic force is given vectorially by

\(\displaystyle \mathbf{F} = q\,(\mathbf{v}\times\mathbf{B})\)

In magnitude form, using the angle θ between v and B,

\(\displaystyle F = q\,v\,B\,\sin\theta\)

Key points: the magnetic force is always perpendicular to both the velocity and the magnetic field; it does no work on the particle (it changes direction, not speed, absent other forces).

Force on a current-carrying conductor

A straight conductor of length L carrying current I in a uniform magnetic field B experiences force

\(\displaystyle F = B\,I\,L\,\sin\theta\)

This is a direct consequence of summing the Lorentz forces on moving charges that constitute the current. The direction follows the right-hand rule (or using the vector form \(\mathbf{F} = I\,\mathbf{L}\times\mathbf{B}\) where \(\mathbf{L}\) is a vector along the conductor with magnitude L).

Vector cross product (3D) method

For robust computations, especially when velocity or field are not perpendicular, use the cross product. If v = (v_x,v_y,v_z) and B = (B_x,B_y,B_z), then

\(\displaystyle \mathbf{v}\times\mathbf{B} = (v_y B_z - v_z B_y,\; v_z B_x - v_x B_z,\; v_x B_y - v_y B_x)\)

Multiply by charge q to get force components. The calculator performs this operation in the Vector tab and returns components and magnitude.

Solenoid field approximation

An ideal long solenoid produces an approximately uniform axial magnetic field given by

\(\displaystyle B = \mu_0\, n\, I\)

where μ₀ ≈ 4π×10⁻⁷ T·m/A is the vacuum permeability, n is number of turns per meter, and I is current. This formula is useful for designing coils and estimating fields in laboratory setups; end effects and finite-length corrections are not included in the simple expression.

Units and practical tips

Use SI units consistently: charge in coulombs (C), velocity in m/s, magnetic field in tesla (T), current in ampere (A), length in meters (m), and force in newton (N). For many laboratory magnets B is measured in mT (millitesla); convert to tesla (1 mT = 1×10⁻³ T) before calculation. Use degrees for intuitive angle entry — the calculator converts to radians internally.

Worked examples

Example 1 — Moving charge: An electron (q = −1.602×10⁻¹⁹ C) moves at v = 2.0×10⁶ m/s perpendicular to a 0.1 T field. Magnitude F = |q| v B = 1.602e-19 × 2.0e6 × 0.1 = 3.204e-14 N. Direction is given by v × B and sign of q.

Example 2 — Conductor: A 0.5 m wire carrying 3 A in a 0.2 T field perpendicular to the wire experiences F = B I L = 0.2 × 3 × 0.5 = 0.3 N.

Example 3 — Vector + solenoid: If you compute B from a solenoid with n = 1000 turns/m and I = 0.5 A, B ≈ μ₀ n I = 4π×10⁻⁷ × 1000 × 0.5 ≈ 6.283×10⁻4 T (0.628 mT). Use that vector in the Lorentz cross-product to compute force on moving charges.

Limitations

The simple formulas assume uniform B and steady currents; real magnets and conductors may need finite-element analysis for high-precision design. The solenoid expression assumes a long solenoid; short coils require correction factors. Also remember magnetic forces do not change kinetic energy of a single charged particle (they do no work on the particle's speed), but they can do work on current distributions when the geometry changes.

Use this calculator for quick checks, classroom problems, lab estimates and for generating clear step-by-step derivations for reports. When designing real hardware, validate with measurements or detailed simulations.