Magnetic Force on a Current-Carrying Wire Calculator
Enter the magnetic field strength, current, wire length, and the angle between the wire and the field to find the Lorentz force acting on the wire. The formula is F = B I L sin(theta). Switch between metric and imperial length units, use degrees or radians, or flip the mode to solve for any unknown variable.
Formula
Worked example
A wire carries 2 A through a 0.5 T field over a 20 cm (0.2 m) segment at 90 degrees. F = 0.5 x 2 x 0.2 x sin(90) = 0.5 x 2 x 0.2 x 1 = 0.2 N. At 30 degrees the same setup gives F = 0.5 x 2 x 0.2 x 0.5 = 0.1 N, exactly half the perpendicular value.
What is the magnetic force on a current-carrying wire?
When an electric current flows through a conductor inside a magnetic field, each moving charge experiences a Lorentz force. The net effect on the wire is given by F = B I L sin(theta), where B is the magnetic flux density in Tesla, I is the current in Amperes, L is the length of wire inside the field in metres, and theta is the angle between the direction of current flow and the direction of the field. The force is perpendicular to both the current and the field direction, following the right-hand rule. This principle is the basis for electric motors, galvanometers, loudspeakers, and rail guns.
How to use this calculator
Select what you want to solve for: force, magnetic field strength, current, or wire length. Enter the known values in the remaining fields, choose your preferred unit for length (metres, centimetres, feet, or inches) and angle (degrees or radians), and the result appears instantly. The "show your work" panel below the result traces every arithmetic step with your actual numbers. The force vs. angle chart shows how the force varies as you rotate the wire from 0 to 180 degrees, and your current angle is marked on the curve.
The physics behind the formula
Each conduction electron moving with drift velocity v experiences a force F = qv x B (the cross product of velocity and field). For a wire carrying current I = nqvA (where n is the charge carrier density and A is the cross-sectional area), the total force on a segment of length L is F = IL x B, whose magnitude is F = BIL sin(theta). The factor sin(theta) captures the cross-product geometry: only the component of the field perpendicular to the current contributes to the force. At theta = 90 degrees the full field acts; at theta = 0 or 180 degrees the field is parallel to the current and the force is zero.
Practical applications
Electric motors exploit this force by running current through coils in a magnetic field; the resulting torque rotates the rotor. Galvanometers use a spring-balanced coil to measure current by how far the deflection goes. Loudspeakers attach a coil to a cone and drive audio-frequency currents through a permanent magnet field, converting electrical signals to sound. In experimental physics, the force between parallel wires carrying current was the original definition of the Ampere, and it underpins precision force standards. Rail guns accelerate a conducting projectile using the same principle with very high currents and fields.
Force as a fraction of maximum at common angles
| Angle (deg) | sin(theta) | % of maximum force |
|---|---|---|
| 0 | 0.0000 | 0% |
| 15 | 0.2588 | 25.9% |
| 30 | 0.5000 | 50.0% |
| 45 | 0.7071 | 70.7% |
| 60 | 0.8660 | 86.6% |
| 75 | 0.9659 | 96.6% |
| 90 | 1.0000 | 100% (maximum) |
| 105 | 0.9659 | 96.6% |
| 120 | 0.8660 | 86.6% |
| 135 | 0.7071 | 70.7% |
| 150 | 0.5000 | 50.0% |
| 165 | 0.2588 | 25.9% |
| 180 | 0.0000 | 0% |
F = B I L sin(theta). Maximum force occurs at 90 degrees. At 0 and 180 degrees, sin(theta) = 0 and the force is zero.
Frequently asked questions
What angle gives the maximum magnetic force?
The maximum force is at theta = 90 degrees (the wire is perpendicular to the magnetic field), where sin(theta) = 1 and F = BIL. At 0 or 180 degrees the wire is parallel to the field, sin(theta) = 0, and the force is zero. Any angle in between gives a force proportional to sin(theta).
Why does the formula use sin(theta) rather than cos(theta)?
The force is the cross product of the current direction vector and the magnetic field vector: F = IL x B. The magnitude of a cross product is |A||B|sin(theta), where theta is the angle between the two vectors. The sine picks out the perpendicular component of the field relative to the current, which is the component that drives a sideways force. Cosine would give the parallel component, which contributes nothing to the lateral force.
What is the direction of the force?
The force direction follows the right-hand rule: point your fingers in the direction of current flow, curl them toward the magnetic field, and your thumb points in the direction of the force on the wire. Reversing the current or the field direction reverses the force. The force is always perpendicular to both the current and the field - it cannot accelerate the wire along its own length.
Does the formula work for a curved wire?
For a straight wire in a uniform field, F = BIL sin(theta) applies directly. For a curved wire or a non-uniform field, you integrate the force along the wire: F = integral of I (dL x B). This calculator handles the straight-wire uniform-field case, which covers most practical engineering problems such as motor windings and force standards.
How does wire resistance affect the magnetic force?
It does not appear in the force formula directly, but resistance determines how much current a given voltage can drive: I = V/R. Higher resistance means less current for the same voltage, which means less force. Resistance also causes I-squared-R heating, which can limit the maximum safe current and therefore the maximum achievable force in continuous operation.
What units must I use for the inputs?
Internally the calculator always uses SI units (Tesla for B, Amperes for I, metres for L, and Newtons for F). If you select centimetres, feet, or inches for the wire length, the calculator converts to metres before computing. The angle can be in degrees or radians - choose whichever your data is in and the calculator converts automatically.