Ampere's Law — Straight Wire

The magnetic field at perpendicular distance r from a very long straight wire carrying steady current I is one of the cornerstone results of magnetostatics: B = μ₀I/(2πr). Field lines are concentric circles centered on the wire, lying in planes perpendicular to the current. The field falls off as 1/r — slower than electrostatic 1/r² because field lines close in loops without beginning or ending on magnetic charges (there are no magnetic monopoles).

μ₀ = 4π × 10⁻⁷ T·m/A is the permeability of free space. Numerical check: I = 10 A at r = 0.05 m gives B = (4π×10⁻⁷)(10)/(2π×0.05) = 4×10⁻⁵ T = 40 μT — comparable to Earth's field but from a lab current.

Direction uses the right-hand grip rule: thumb along current, curled fingers show B direction around the wire. Reversing current reverses the sense of circulation.

Ampère's circuital law ∮ B⃗·dℓ⃗ = μ₀I_enc generalizes this: choose a circular Amperian loop of radius r around the wire; symmetry gives B constant on the loop, so B(2πr) = μ₀I.

The formula assumes the wire is much longer than r so end effects are negligible. For finite wires, use the Biot–Savart law and integrate. Inside a thick wire with uniform current density, B grows linearly with r inside and falls as 1/r outside.

Applications include estimating fields near power lines, designing electromagnets (as building block for solenoids), and explaining why two parallel wires exert forces on each other.