--- title: "Ampere's Law: Wires, Solenoids and Toroids" description: "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" canonical: https://lightmysky.com/learn/science/amperes-law-wires-solenoids-and-toroids-mt_uTzRcSiUQ- source: https://lightmysky.com/learn/science/amperes-law-wires-solenoids-and-toroids-mt_uTzRcSiUQ-.md retrieved: 2026-09-12 --- > **Agent view.** This is the Markdown twin of the page, for tools and assistants. > When to use this site, and the call that answers each job: https://lightmysky.com/agent-instructions.md > API description (OpenAPI 3.1): https://lightmysky.com/openapi.json · Authentication: https://lightmysky.com/auth.md > Pricing: https://lightmysky.com/pricing.md · Catalog: https://lightmysky.com/llms.txt · Full catalog: https://lightmysky.com/llms-full.txt > Every machine-readable file on this domain: https://lightmysky.com/.well-known/ai-catalog.json > Ask for Markdown with `Accept: text/markdown`, a `.md` address, or `?mode=agent`. # 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. Subject: Science · Area: Electricity & Magnetism · Ages 19 to 21 Page: https://lightmysky.com/learn/science/amperes-law-wires-solenoids-and-toroids-mt_uTzRcSiUQ- ## Ready when they can - 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 ## Lesson: Ampere loops for symmetric currents 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. **Example.** 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. **Tip.** 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. **Recap.** Ampere's law turns symmetric currents into one-line fields, and without symmetry the integral cannot be unpacked. ## Practice 8 questions on this page, each with its working shown. ## Needs first - [The Biot-Savart Law and the Field of a Current Element](https://lightmysky.com/learn/science/the-biot-savart-law-and-the-field-of-a-current-element-mt_RRsHUaXrFq) ## Opens up - [Magnetisation: Diamagnetism, Paramagnetism and Ferromagnetism](https://lightmysky.com/learn/science/magnetisation-diamagnetism-paramagnetism-and-ferromagnetism-mt_1apyQ1_1gg) - [Induced Electric Fields and Motional EMF](https://lightmysky.com/learn/science/induced-electric-fields-and-motional-emf-mt_LpmHnz46tu) - [Displacement Current and Maxwell's Equations](https://lightmysky.com/learn/science/displacement-current-and-maxwells-equations-mt_YR8m8TNfyy)