Ampere's Law: Wires, Solenoids and Toroids

Biot-Savart adds fields from current elements. For steady currents in vacuum, Ampere's law instead relates circulation to current: integral around a closed path of B dot dl = mu0 Ienc. The dot product selects the field component along each path element. Ienc is the signed current crossing a surface bounded by the path. This equation gives circulation, not automatically the field at each point.

Curl your right fingers in the chosen path direction; your thumb gives the positive surface normal. Crossings with it count positive and crossings against it count negative. For example, +4 A and -1 A give Ienc = 3 A. External currents can affect B on the path, although their contribution to its circulation is zero. Zero net enclosed current does not by itself mean zero local field.

For an infinite straight wire, choose a circle centred on the wire in a perpendicular plane. Cylindrical symmetry makes B tangent to the circle and constant in magnitude. Thus integral B dot dl = B(2 pi r) = mu0 I, giving B = mu0 I/(2 pi r). This agrees with Biot-Savart. A long real wire approximates this result away from its ends.

An ideal infinite solenoid has uniformly distributed turns, with vacuum inside. Symmetry and the condition that the field vanishes far outside give zero exterior field and a uniform axial interior field. A finite, tightly wound solenoid approximates this near its middle when its length greatly exceeds its radius. Its exterior field need not be zero.

These bars compare magnitudes in the ideal model; they do not draw field lines.

For that ideal solenoid, use a rectangle with an axial side of length l inside and a matching side outside. The outside field is zero; the end segments are perpendicular to B. Only the inside side contributes B l. With n = N/L turns per metre, the surface crosses n l turns, so B l = mu0 n l I and B = mu0 n I. At fixed model conditions, field magnitude scales with n times I.

For an ideal toroid, N circular turns are spread uniformly around a ring with vacuum in its winding region. At radius r from its central axis within that region, symmetry makes B tangent and constant on a centred circle. Its surface crosses N turns: B(2 pi r) = mu0 N I, so B = mu0 N I/(2 pi r). Symmetry also gives zero field in the hole and exterior. Real windings and leads can produce leakage.

An infinite circular wire of radius a carries total current I uniformly across its cross-section. A centred circle of radius r <= a encloses current in proportion to area: Ienc/I = pi r^2/(pi a^2) = (r/a)^2. Then B(2 pi r) = mu0 I(r/a)^2, so B = mu0 I r/(2 pi a^2) inside. Outside, all I is enclosed and B = mu0 I/(2 pi r).

The bars show current shares, not a drawing of the wire's cross-section.

Without enough symmetry, Ampere's law still determines circulation from signed current. Finding a local B directly requires a path where its tangential component and the other contributions are known. A single circular current loop has rotational symmetry, but no useful path isolates its axial field as B times a known length. Use the Biot-Savart axial calculation for that case. The circulation law still applies.