Last updated: 2026-06-15

Magnetic Field of a Long Straight Wire

A current-carrying wire generates a magnetic field in the space around it. For a straight wire that is long compared with the distances involved, the field at a perpendicular distance $r$ from the wire is:

$B = \frac{\mu_0 I}{2\pi r}$

where $I$ is the current in amperes, $r$ is the distance in metres, and $\mu_0 = 1.2566 \times 10^{-6}\ \text{T·m/A}$ is the permeability of free space. This result follows from both the Biot–Savart law and Ampère's law; the two approaches agree exactly for an infinite straight wire.

Direction of the Field

The magnetic field forms concentric circles centred on the wire. The right-hand rule gives the direction: point your right thumb in the direction of the conventional current, and your curled fingers show the direction the field lines rotate around the wire. At any point the field is tangential to the circle passing through that point — perpendicular to both the wire axis and the radial direction.

Formula

The field falls off as $1/r$: doubling the distance halves the field. This linear inverse law contrasts with the inverse-square law for point charges because the geometry of an infinite wire is cylindrical rather than spherical.

Worked Example

A wire carries a current of 10 A. Find the magnetic field strength at a perpendicular distance of 5 cm (0.05 m).

Entering 10 A and 5 cm in the calculator gives the same result: approximately 40 µT. For context, Earth's surface magnetic field is roughly 25–65 µT, so a 10 A wire at 5 cm produces a field comparable to Earth's own field.

Practical Considerations

In real systems the wire has finite length, and the formula above is an approximation valid when $r$ is much smaller than the wire length. For shorter wires the Biot–Savart law must be integrated numerically over the wire's actual length, producing a smaller field. The approximation is accurate to better than 1 % when the distance is less than about one-tenth of the wire length.

The formula also assumes the wire carries a steady direct current. With alternating current, at high frequencies the skin effect concentrates the current near the wire's surface, which modifies the external field at close range. At the distances and frequencies encountered in most practical calculations the approximation remains valid.

Related Calculations

The magnetic field produced by a straight wire is the basis for understanding more complex geometries. A solenoid — a coil of wire — superimposes the fields of many parallel wires and produces a much stronger, nearly uniform interior field. Parallel wires carrying currents in the same direction attract each other because each wire sits in the other's magnetic field; this is the physical basis for the SI definition of the ampere.