Magnetic Force Calculator (F = BIL and F = qvB)

Magnetic Force Guide

What do I need to know about Force on a Current-Carrying Wire?

F = BIL sin θ. Where B = magnetic flux density (tesla, T), I = current (A), L = length of conductor in the field (m), θ = angle between conductor and field. Maximum force when θ = 90° (conductor perpendicular to field). Zero force when θ = 0° (conductor parallel to field). Example: a 10 cm wire carrying 2A in a 0.5T field at 90°: F = 0.5 × 2 × 0.1 × sin(90°) = 0.1N. Direction: Fleming's left-hand rule — first finger: field direction (north to south), second finger: conventional current direction, thumb: resulting force direction.

What should I know about Force on a Moving Charge?

F = qvB sin θ. Where q = charge (C), v = velocity (m/s), B = magnetic flux density (T), θ = angle between velocity and field. An electron (q = 1.6×10⁻¹⁹ C) moving at 1×10⁶ m/s perpendicular to a 0.5T field: F = 1.6×10⁻¹⁹ × 10⁶ × 0.5 = 8×10⁻¹⁴ N. This tiny force is huge compared to the electron's weight (9.1×10⁻³¹ kg × 9.81 = 8.9×10⁻³⁰ N). Magnetic force does no work on a charge — it only changes direction, never speed. This is why charged particles move in circles (not spirals) in uniform magnetic fields, with the field strength setting the radius of the circular path.

What do I need to know about Circular Motion in Magnetic Field?

The magnetic force provides the centripetal force: qvB = mv²/r. Therefore radius r = mv/(qB). Larger mass, higher speed, weaker field → larger radius. Larger charge, stronger field → smaller radius. A proton (m=1.67×10⁻²⁷ kg, q=1.6×10⁻¹⁹ C) at 10⁷ m/s in a 1T field: r = (1.67×10⁻²⁷ × 10⁷)/(1.6×10⁻¹⁹ × 1) = 0.104m. The cyclotron (particle accelerator) uses this principle — particles spiral outward as they are accelerated, with the radius increasing as speed increases.

What should I know about Applications?

Electric motors: F=BIL force on current-carrying coils in a magnetic field produces rotation. The motor effect. Loudspeakers: current through a coil in a magnetic field creates oscillating force, vibrating the cone. Cathode ray tubes (oscilloscopes, old TVs): electron beam deflected by magnetic force for scanning. Mass spectrometer: ions of different mass have different radii in the same field — allows separation and identification of isotopes. MRI scanners: strong magnetic fields, typically 1.5-3 Tesla, align hydrogen nuclei in the body's tissue, which is then used to build detailed internal images without ionising radiation.

What should I know about the Force on a Current-Carrying Wire Formula?

The force on a current-carrying wire in a magnetic field is F = BIL×sin(θ), where B is magnetic flux density (tesla), I is current (amps), L is wire length in the field (metres), and θ is the angle between the wire and the field direction. Force is maximum (F=BIL) when the wire is perpendicular to the field (θ=90°) and zero when parallel to the field (θ=0°, sin(0)=0) — a wire running exactly along the field lines experiences no magnetic force regardless of how much current flows. For a 0.5m wire carrying 2A perpendicular to a 0.3T field: F = 0.3×2×0.5×sin(90°) = 0.3N. This formula is the basis of every electric motor: current-carrying coils positioned within a magnetic field experience this force, and arranging multiple coils to rotate continuously converts electrical energy into mechanical rotation.

What do I need to know about the Right-Hand Rule for Direction?

The direction of magnetic force is found using the right-hand rule (for conventional current and the F=BIL or F=qvB formulas): point fingers in the direction of current flow (or positive charge velocity), curl them toward the magnetic field direction, and the thumb points in the direction of the resulting force — or more precisely for F=IL×B, point fingers along current, then curl toward B, and the force follows the cross-product direction perpendicular to both. This is why the force on a moving charge or current is always perpendicular to both the velocity/current and the field, never parallel to either — a fundamentally different geometry from electric or gravitational forces, which act directly along the line between source and object. This perpendicularity is exactly why a charged particle moving through a uniform magnetic field doesn't speed up or slow down (the force does no work, since it's always perpendicular to velocity) but instead curves into a circular path.

What do I need to know about the Force Between Two Parallel Wires?

Two long, straight, parallel current-carrying wires exert a force on each other per unit length of F/L = μ₀I₁I₂/(2πd), where d is the separation between them. The force is attractive when the currents flow in the same direction and repulsive when they flow in opposite directions — the opposite of the rule for like and unlike electric charges. This wire-on-wire force was historically so fundamental and precisely measurable that, before 2019, it formed the official SI definition of the ampere itself.

Magnetic Field Strength Reference Table

Magnetic flux density (B, measured in teslas) spans an enormous range in nature and technology:

The jump from a fridge magnet to an MRI scanner is already a factor of roughly 150-3,000×, and a neutron star's field is roughly ten billion times stronger again than the strongest magnets humans can currently build in a lab.

Worked Example: Force on a Wire in Earth's Magnetic Field

A straight horizontal wire 2 m long carries a current of 15 A and is oriented perpendicular to Earth's magnetic field, which has a strength of about B = 5 × 10⁻⁵ T at the surface. Find the magnetic force on the wire. Using F = BIL sin(θ), with θ = 90° (perpendicular, so sin θ = 1): F = (5 × 10⁻⁵ T) × (15 A) × (2 m) × 1 = 5 × 10⁻⁵ × 30 = 1.5 × 10⁻³ N = 0.0015 N (1.5 mN). That's an extremely small force — comparable to the weight of about 0.15 grams — which illustrates why Earth's own magnetic field, despite being easily detectable with a compass, exerts negligible mechanical force on ordinary current-carrying wires and household electronics. By contrast, placing that same wire inside an MRI scanner's 3 T field (60,000 times stronger than Earth's) with the same 15 A current would produce F = 3 × 15 × 2 = 90 N — strong enough to visibly deflect or even snap an unsecured wire, which is exactly why metal objects and loose cables are kept away from MRI machines.

Worked Example: Radius of Circular Motion for a Charged Particle

An electron (charge q = 1.6 × 10⁻¹⁹ C, mass m = 9.11 × 10⁻³¹ kg) moves at v = 2 × 10⁶ m/s perpendicular to a magnetic field of B = 0.5 T. Find the radius of the circular path it follows. The magnetic force provides the centripetal force for circular motion: qvB = mv²/r, which rearranges to r = mv/(qB). r = (9.11 × 10⁻³¹ × 2 × 10⁶) / (1.6 × 10⁻¹⁹ × 0.5) = (1.822 × 10⁻²⁴) / (8 × 10⁻²⁰) = 2.2775 × 10⁻⁵ m, or about 22.8 micrometers. This tiny radius — thinner than a human hair — is exactly why particle accelerators and mass spectrometers need extremely strong magnetic fields or very high particle speeds to produce circular paths large enough to be measured and controlled practically. Doubling the field strength to B = 1 T would halve the radius to about 11.4 μm, while doubling the particle's speed would double the radius back to about 45.6 μm — a direct consequence of r being proportional to v but inversely proportional to B. This same r = mv/(qB) relationship, rearranged to p = mv = qBr, is exactly how particle physicists infer a particle's momentum from the curvature of its track in a bubble chamber or modern silicon tracker without needing to measure its speed directly — a tighter curve means lower momentum, and a straighter track means higher momentum, for a fixed known field strength. This is also the basic principle behind a mass spectrometer, which separates ions of different mass-to-charge ratios by measuring how tightly each one curves in a known magnetic field, then matching the measured radius back to the elements or isotopes it corresponds to.