Magnetic Force Calculator | Lorentz

The magnetic force is F = q(v × B), a cross product. The cross product of two vectors is always perpendicular to both.

• ›This means the force is always perpendicular to the velocity vector.
• ›Therefore, the magnetic force can never do work on a charge (work requires a component of force parallel to displacement).
• ›The magnetic force changes the direction of motion but never the speed or kinetic energy.
• ›This is why magnetic fields are used to steer charged particles in accelerators without accelerating them.

For a positive charge moving with velocity v in field B:

• ›Point your right-hand fingers in the direction of v (velocity).
• ›Curl your fingers toward B (magnetic field direction).
• ›Your thumb points in the direction of force F on the positive charge.
• ›For a negative charge (e.g., electron), the force is in the opposite direction, use your left hand, or flip the result.

For a wire: point fingers in the direction of conventional current I, curl toward B, thumb gives force direction (Fleming's left-hand rule in UK notation).

The magnetic force F = qvB sin(θ) is zero when sin(θ) = 0, which occurs when:

• ›θ = 0°, the velocity (or current) is parallel to the magnetic field.
• ›θ = 180°, the velocity is anti-parallel to the field.
• ›v = 0, a stationary charge experiences no magnetic force (only electric force if E ≠ 0).
• ›q = 0, a neutral particle experiences no magnetic force.

Maximum force occurs at θ = 90° (perpendicular), where sin(90°) = 1.

When a charged particle moves perpendicular to a uniform magnetic field, the constant perpendicular Lorentz force curves its path into a circle:

• ›Radius: r = mv / (qB), larger for faster or heavier particles, smaller for larger B or charge.
• ›Cyclotron frequency: f = qB / (2πm), constant regardless of speed.
• ›Period: T = 2πm / (qB), independent of speed (key to cyclotron operation).

Cyclotrons exploit this: the particle spirals outward as it gains energy but completes each orbit in the same time, allowing synchronised radio-frequency acceleration.

Each wire creates a magnetic field around itself. The other wire sits in that field and carries a current, so it experiences a force.

• ›Wire 1 creates a B field circling around it (by the right-hand rule).
• ›At the location of Wire 2, Wire 1's field points in a direction such that F = I₂L × B₁ points toward Wire 1.
• ›The same happens symmetrically, Wire 2's field pulls Wire 1 toward Wire 2.
• ›Result: same-direction currents attract.

For opposite currents, the field directions reverse, making the forces repulsive. This is the physical basis of the original SI definition of the Ampere.

The SI unit is the Tesla (T): 1 T = 1 kg/(A·s²) = 1 V·s/m².

• ›1 Tesla = 10,000 Gauss (the CGS unit)
• ›Earth's magnetic field: ~25–65 μT (microtesla)
• ›Typical fridge magnet: ~1–5 mT
• ›MRI scanner: 1.5 T to 3 T (clinical), up to 10.5 T (research)
• ›Strongest sustained superconducting magnet: ~45 T
• ›Neutron star surface: ~10⁸ to 10¹¹ T

An electric motor works by placing current-carrying conductors in a magnetic field. The Lorentz force F = BIL sin(θ) on the conductors creates a torque that rotates the shaft.

• ›The torque τ = nBIA sin(θ), where n = number of turns, A = coil area.
• ›Maximum torque at θ = 90°, the plane of the coil parallel to the field.
• ›A commutator (in DC motors) or AC supply (in AC motors) reverses the current at the right moment to maintain torque in one rotational direction.
• ›Generator operation is the reverse: mechanical rotation induces current via electromagnetic induction.

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