Ampère's circuital Law

Ampère's circuital law, with a correction due to Maxwell, relates the circulation of a magnetic field around an area to the Current density passing through it:

\oint 𝜕 Ω ⃗ 𝐁 \cdot 𝑑 ⃗ ℓ = 𝜇 0 (\iint Σ ⃗ 𝐉 \cdot 𝑑 ⃗ 𝐚 + 𝜖 0 𝑑 𝑑 𝑡 \iint Σ ⃗ 𝐄 \cdot 𝑑 ⃗ 𝐚) = 𝜇 0 (𝐼 Σ + 𝜖 0 𝜕 Φ 𝐸, Σ 𝜕 𝑡)

or equivalently by Stokes's theorem

⃗ \nabla \times ⃗ 𝐁 = 𝜇 0 (⃗ 𝐉 + 𝜖 0 𝜕 ⃗ 𝐄 𝜕 𝑡)