Ampere's circuital law

Ampere's circuital law is to magnetism what Gauss's law is to electricity: a short cut when the current is arranged symmetrically.

∮ B · dl = µ₀ I_enclosed: add up B along a closed loop, and you get µ₀ times the current passing through that loop.

Choose a loop (an Amperian loop) on which B is the same size and runs along the loop. Then the left side is just B × (length of the loop).

Long straight wire: take a circle of radius r round it. B × 2πr = µ₀I, so:

B = µ₀I / (2πr). Example: 10 A at 5 cm gives B = 2 × 10⁻⁷ × 10 / 0.05 = 4 × 10⁻⁵ T.

The field lines are circles around the wire. Right-hand thumb rule: thumb along the current; the curled fingers show the way B goes round.

Inside a thick wire of radius a with current spread evenly, the loop encloses only part of the current, and B = µ₀Ir / (2πa²). B grows from zero at the centre to its largest at the surface, then falls as 1/r outside.

Long solenoid (a coil wound like a spring), with n turns per metre:

B = µ₀ n I inside, uniform, and almost zero outside.

Example: 1000 turns per metre carrying 2 A: B = 4π × 10⁻⁷ × 1000 × 2 ≈ 2.5 × 10⁻³ T. The field depends on turns per metre and the current, not on the solenoid's width.