Ampere's Law: Wires, Solenoids and Toroids
The line integral of the magnetic field around a closed loop equals the enclosed current, which gives the field of symmetric arrangements in a line of algebra. A long solenoid comes out uniform inside and near zero outside.
What a learner can do afterwards
- States Ampere's law and chooses a loop that matches the symmetry of the current
- Derives the field of a long straight wire, a solenoid and a toroid
- Says why the law gives nothing useful without symmetry
1 · Read
Ampere's law says the loop integral of B around any closed path equals mu0 times the current passing through it. Curl your right fingers along the path and your thumb marks positive current. Pick a loop that matches the symmetry: a circle around a straight wire, or a rectangle straddling a solenoid wall.
Around a straight wire, B is constant on a circle of radius r, so B times 2 pi r equals mu0 I, giving B equals mu0 I over 2 pi r. A rectangle crossing a solenoid wall gives B times l equals mu0 n l I, so inside B equals mu0 n I with n turns per length, and nearly zero outside. A 300-turn, 14.0 cm solenoid at 0.410 A has n of 2140 per metre and about 0.00110 T inside. A toroid traps B equals mu0 N I over 2 pi r in its core and zero outside.
The law only hands you an integral, so without symmetry it says little. On a symmetric loop B has one size and runs along the path, so it factors out and you solve in one line. A short finite wire or a single loop has no such loop, which is why those need the Biot-Savart sum instead. Inside a uniform thick wire the enclosed current grows with r squared, so B grows linearly with r.
Count only the current your loop encloses, with sign from the right-hand rule. Current outside the loop adds nothing. Never pull B out of the integral unless symmetry makes it constant along your chosen path.
Ampere's law turns symmetric currents into one-line fields, and without symmetry the integral cannot be unpacked.
2 · Watch
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Where it sits
8 questions wait behind this lesson, each with its answer explained. Every answer feeds the sky: stars light as they are learned, and dim when it is time to come back.