Ampere’s circuital law and its uses
Ampere’s circuital law: the line integral of B around any closed loop equals μ₀ times the net current passing through the loop: ∮ B · dl = μ₀ I_enclosed The closed path is called an Amperian loop. Like Gauss’s theorem in electrostatics, it is most useful when there is symmetry, so that B is constant along the path.
Long straight wire: take a circle of radius r around the wire. B is the same everywhere on it and along the circle, so B × 2πr = μ₀I, giving B = μ₀I / (2πr). The field lines are concentric circles; direction by the right-hand thumb rule (thumb along current, fingers curl along B).
Worked example: I = 10 A, r = 0.05 m: B = 2 × 10⁻⁷ × 10 / 0.05 = 4 × 10⁻⁵ T.
Long solenoid: a long coil with n turns per unit length carrying current I. Inside, the field is uniform and along the axis: B = μ₀ n I. Outside a long solenoid the field is nearly zero. At the ends the field is about half the value at the centre.
Worked example: n = 1000 turns/m, I = 2 A: B = 4π × 10⁻⁷ × 1000 × 2 ≈ 2.5 × 10⁻³ T.
Toroid: a solenoid bent into a ring; B = μ₀NI/(2πr) inside the core, zero outside and in the open space.
Key points
- ∮B · dl = μ₀I_enclosed.
- Straight wire: B = μ₀I/2πr (∝ 1/r).
- Solenoid: B = μ₀nI, where n = N/L.
- Field inside an ideal solenoid is uniform; outside it is almost zero.
Exam tips
- Always convert total turns to turns per metre for solenoids.
- Know the inside/outside results for a toroid and solenoid.
Common mistakes
- Using total N instead of n = N/L in μ₀nI.
- Using 1/r² for a straight wire (it is 1/r).
- Including currents outside the Amperian loop.