--- title: "Ampere's Law: Wires, Solenoids and Toroids" description: "For steady currents in vacuum, the closed-path integral of B dot dl equals mu0 times signed enclosed current. Sufficient symmetry lets this determine fields. An ideal infinite solenoid has a uniform i" 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: 2026-11-12 --- > **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 For steady currents in vacuum, the closed-path integral of B dot dl equals mu0 times signed enclosed current. Sufficient symmetry lets this determine fields. An ideal infinite solenoid has a uniform interior field and zero exterior field. 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 - Explains why Ampere's law constrains circulation but usually cannot determine a local field without sufficient symmetry ## Lesson: Ampere loops for symmetric currents 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. 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). 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. **Recap.** For steady currents in vacuum, circulation equals mu0 times signed enclosed current. Use symmetry to find local fields, and keep the ideal wire, solenoid and toroid assumptions explicit. *Original lesson revisions, questions and flashcards written for Marble. Physics facts and calculation methods were studied using University Physics Volume 2, sections 12.5 and 12.6, and College Physics, section 22.9; textbook prose is not reproduced.* ## 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)