How this instrument works
A wire carrying current through a magnetic field feels a mechanical force, not because the wire itself is magnetic, but because every moving charge inside it does. Each electron drifting along the conductor feels the magnetic half of the Lorentz force individually; sum that force over every carrier in a length L of wire carrying current I, and the messy per-particle sum collapses into one clean expression: F = BIL sinθ. The field B and the current I set the scale, the length L says how much wire is actually exposed to the field, and sinθ decides how much of that exposure counts.
The angle term is doing real physical work, not just trigonometric decoration. Only the component of current running crosswise to the field lines gets pushed sideways; a wire laid exactly along the field, θ = 0°, carries current with nothing to deflect it, and sinθ vanishes to confirm it. Turn the same wire until it runs squarely across the field, θ = 90°, and sinθ reaches its maximum of 1 — every ampere in that segment contributes its full share of force. This is precisely the geometry a motor designer chases: armature windings are shaped and commutated so their current always crosses the field near 90°, because any shallower angle wastes current without producing torque.
This exact relation is more than a classroom identity — it is the working equation behind a loudspeaker's 'BL product' (force factor, in tesla-metres, printed on driver datasheets to predict cone force from drive current) and behind the mechanical bracing calculations engineers run on parallel busbars, where a short-circuit current spike can turn ordinary copper bars into projectiles for a few milliseconds. The formula assumes a straight wire segment sitting in a magnetic field that is uniform across its length; a curved conductor or a field that varies along the wire needs the differential form, dF = I dL × B, integrated piece by piece instead of one multiplication.
- Set Magnetic field strength, T to the field's flux density in tesla — there is no unit menu here, so convert gauss first (1 T = 10,000 G).
- Enter Wire current in amperes — the steady current actually flowing through the segment, as read off a meter or nameplate.
- Enter Wire length in the field in metres or centimetres — only the portion of wire actually inside the field counts, not the conductor's full length.
- Set Angle between wire and field in degrees: 90° for a wire running straight across the field, 0° for a wire running along it.
- Read Force on the wire in newtons — the mechanical push that segment feels, ready to compare against a bracket, mount, or bearing rating.
Worked example — 3 A across 2 m, square to a 0.5 T field
A straight 2 m segment of wire carries 3 A and sits inside a 0.5 T field, oriented exactly perpendicular so the angle between wire and field reads 90°. Since sin 90° = 1, none of the current's push is lost to geometry: F = 0.5 × 3 × 2 × 1 = 3 N. That reading is exactly what a motor designer aims for at every winding — conductors wound to cross the field squarely, because any shallower angle would produce less force from the same 3 A.
Rotate that same 2 m, 3 A wire until it runs parallel to the field instead — angle at 0° — and the force drops to zero, because sin 0° = 0; current can flow directly along field lines all day without feeling a sideways push. Hold the wire perpendicular again but double the field to 1 T, and the same segment now reads 6 N, exactly twice the original 3 N, since force scales linearly with field strength once the angle is fixed.
Questions
Why does the sine of the angle appear in the formula?
Because only the slice of current running crosswise to the field actually gets pushed sideways. Decompose the current into a component along the field and a component across it — the along-field piece feels nothing, and the across-field piece is exactly I sinθ. Multiply that effective current by B and L and sinθ falls straight out of the geometry, not as an approximation but as an exact consequence of how the underlying Lorentz force depends on the angle between velocity and field.
How does this formula relate to the Lorentz force on a single charge?
It is the same force, added up. A wire's current is really countless charge carriers drifting together, each one feeling qvB sinθ from the field individually. Sum that push over every carrier packed into a length L carrying current I, and the individual charge, drift speed, and carrier count fold into one measurable quantity, current, collapsing the sum to F = BIL sinθ. The wire version is the practical, meter-readable shortcut; the single-charge version is what is actually happening underneath.
What if the wire is curved or the field is not uniform?
Then this formula only applies piece by piece. F = BIL sinθ assumes a straight segment sitting in a magnetic field that does not change across its length; a curved conductor, or a field that strengthens or weakens along the wire, needs the differential version, dF = I dL × B, integrated over the whole path. For a short, straight segment in a roughly uniform field — a motor winding, a busbar section — the simple multiplication is accurate enough for real engineering work.
Why do heavy-current busbars need mechanical bracing against short-circuit forces?
Because the force in F = BIL sinθ scales with current, and a short circuit can push current to ten or more times a busbar's rated value. Since the field a nearby conductor sees also rises with current, the electrodynamic force on a busbar during a fault can spike to roughly a hundred times its everyday value for the brief duration of the fault, enough to bend or shear inadequately braced copper. Switchgear designers size bracing specifically against this transient spike, not the modest force present at normal load.
What is the 'BL product' printed on a loudspeaker driver's datasheet?
It is the B and L from this exact formula, multiplied together and quoted as one number in tesla-metres. A driver's voice coil sits in a fixed magnetic gap of field B, wound with a fixed length L of wire, so for that speaker BL stays constant; multiply it by the drive current from the amplifier and F = (BL) × I gives the force pushing the cone, without needing B and L listed separately. A higher BL product means more force, and more control over the cone, per amp of drive current.
Which way does the wire actually get pushed?
Perpendicular to both the current and the field at once, following the right-hand rule: point the fingers of a right hand along the current, curl them toward the field, and the thumb gives the force direction. (Motor textbooks often show the same construction as Fleming's left-hand rule, built with the opposite hand convention.) This calculator returns only the size of that push in newtons; sketching current and field as vectors is still the fastest way to pin down which way the wire actually moves.