--- title: "The Biot-Savart Law and the Field of a Current Element" description: "Each short piece of current makes a magnetic field that circles it, falling off with the square of the distance, and the total field is the integral of those contributions. It is the magnetic counterp" canonical: https://lightmysky.com/learn/science/the-biot-savart-law-and-the-field-of-a-current-element-mt_RRsHUaXrFq source: https://lightmysky.com/learn/science/the-biot-savart-law-and-the-field-of-a-current-element-mt_RRsHUaXrFq.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`. # The Biot-Savart Law and the Field of a Current Element Each short piece of current makes a magnetic field that circles it, falling off with the square of the distance, and the total field is the integral of those contributions. It is the magnetic counterpart of the Coulomb integral. Subject: Science · Area: Electricity & Magnetism · Ages 19 to 20 Page: https://lightmysky.com/learn/science/the-biot-savart-law-and-the-field-of-a-current-element-mt_RRsHUaXrFq ## Ready when they can - Applies the Biot-Savart law to find the field of a straight wire and on the axis of a loop - Uses the cross product to fix the direction of each contribution - Finds the force between two parallel currents and says when it is attractive ## Lesson: Adding up magnetism piece by piece Every short piece of current makes its own magnetic field that circles the piece. The Biot-Savart recipe says each contribution shrinks with the square of the distance and with the sine of the angle, and its direction comes from the cross product of the piece with the radial direction. The full field is the sum, or integral, of all the pieces. **Example.** For a long straight wire the pieces add to circles around the wire with size B equals mu0 I over 2 pi r. At the centre of a circular loop of radius R the sum is simpler: B equals mu0 I over 2 R, pointing along the axis. An arc covering angle theta gives mu0 I theta over 4 pi r, and the full loop is the theta equals 2 pi case. A 1.0 cm piece carrying 2.0 A makes only about 2.0e-9 T one metre away. Two parallel wires feel each other because each sits in the circular field of the other. The force on a length L is I2 L times B1, where B1 equals mu0 I1 over 2 pi r. Currents in the same direction pull toward each other, and currents in opposite directions push apart. **Tip.** Fix direction before size. Point your thumb with the current and your fingers curl the way B circles. For a loop, curl your fingers with the current and your thumb points along the axis. Never use the straight-wire 1 over r result for a short piece: that 1 over r comes from integrating, while each piece falls as 1 over r squared. **Recap.** Each current piece contributes a circling field, integration gives the wire and loop results, and parallel currents attract or repel by direction. ## Practice 8 questions on this page, each with its working shown. ## Needs first - [Kirchhoff's Rules for Multi-Loop Circuits](https://lightmysky.com/learn/science/kirchhoffs-rules-for-multi-loop-circuits-mt_AS3KiGstGD) - [Charged Particles Moving in a Magnetic Field](https://lightmysky.com/learn/science/charged-particles-moving-in-a-magnetic-field-mt_evuhZ6cSfI) - [Magnetic Flux Density and the Force on a Current-Carrying Wire](https://lightmysky.com/learn/science/magnetic-flux-density-and-the-force-on-a-current-carrying-wire-mt_zuqc4gjBVh) ## Opens up - [Ampere's Law: Wires, Solenoids and Toroids](https://lightmysky.com/learn/science/amperes-law-wires-solenoids-and-toroids-mt_uTzRcSiUQ-)