Magnetic field from a long straight wire

An infinitely long conductor with current $𝐼$ results in a magnetic field strength $⃗ 𝐁$ at a perpendicular distance $𝑦$ .¹

While this scenario is impossible for obvious reasons, it approximates the $⃗ 𝐁$ -field short distances from a long section of straight wire in a large circuit. Since $𝑑 ⃗ ℓ$ is always in the direction of current², the right hand rule shows that any point with a positive $𝑦$ in the above setup will experience $⃗ 𝐁$ out of the page (while the opposite is true for negative $𝑦$ ). Using these observations, solving the Biot-Savart Law for this setup gives

∣ ⃗ 𝐁 ∣ = 𝜇 0 𝐼 2 𝜋 𝑦

where the direction of the $⃗ 𝐁$ -field can be determined by a right hand rule: point the thumb in the direction of current and the curl of the fingers corresponds to the direction of the $⃗ 𝐁$ -field.

Combining this with Magnetic force on a wire, we can derive the Force between two straight current carrying wires.

#state/tidy | #SemBr | #lang/en

2020. Magnetism, pp. 25ff. ↩
The derivation of the given form of the Biot-Savart Law defines $𝑑 ⃗ ℓ$ in this way so as to replace $⃗ 𝐈 𝑑 ℓ$ with $𝐼 𝑑 ⃗ ℓ$ . ↩

Magnetic field from a long straight wire

Footnotes