Electric and magnetic potentials

Gauß's law for magnetic flux and Faraday's law of induction are satisfied by the electric field $⃗ 𝐄$ and Magnetic field $⃗ 𝐁$ iff there exists an electric potential $𝑉$ and a magnetic potential $⃗ 𝐀$ such that

⃗ 𝐄 = - ⃗ \nabla 𝑉 - 𝜕 ⃗ 𝐀 𝜕 𝑡 ⃗ 𝐁 = ⃗ \nabla \times ⃗ 𝐀

Proof

Gauß's law for magnetic flux is satisfied iff the $⃗ 𝐁$ -field is incompressible, which is equivalent to the statement that there exists a vector potential $⃗ 𝐀$ with $⃗ 𝐁 = ⃗ \nabla \times ⃗ 𝐀$ . Then inspecting Faraday's law of induction,

⃗ \nabla \times ⃗ 𝐄 + 𝜕 ⃗ 𝐁 𝜕 𝑡 = ⃗ 𝟎 ⟺ ⃗ \nabla \times ⃗ 𝐄 + 𝜕 𝜕 𝑡 (⃗ \nabla \times ⃗ 𝐀) = ⃗ 𝟎 ⟺ ⃗ \nabla \times (⃗ 𝐄 + 𝜕 ⃗ 𝐀 𝜕 𝑡) = ⃗ 𝟎

we find it holds iff $⃗ 𝐄 + 𝜕 ⃗ 𝐀 𝜕 𝑡$ is irrotational, i.e. admits a scalar potential $𝑉$ such that $- ⃗ \nabla 𝑉 = ⃗ 𝐄 + 𝜕 ⃗ 𝐀 𝜕 𝑡$ . Therefore $⃗ 𝐄 = - ⃗ \nabla 𝑉 - 𝜕 ⃗ 𝐀 𝜕 𝑡$ .

The electric and magnetic fields are Gauge invariant under the transformation

⃗ 𝐀 \to ⃗ 𝐀 + ⃗ \nabla 𝑓 𝑉 \to 𝑉 - 𝜕 𝑓 𝜕 𝑡

Proof

Applying the identity $⃗ \nabla \times (⃗ \nabla 𝑓) = 0$ it immediately follows that $⃗ 𝐀' = ⃗ 𝐀 + ⃗ \nabla 𝑓$ gives the same $⃗ 𝐁$ -field as $⃗ 𝐀$ , however if we want $⃗ 𝐄$ to also be the same we require that

- ⃗ \nabla 𝑉 - 𝜕 ⃗ 𝐀 𝜕 𝑡 = - ⃗ \nabla 𝑉' - 𝜕 𝜕 𝑡 (⃗ 𝐀 + ⃗ \nabla 𝑓) = - \nabla (𝑉' + 𝜕 𝑓 𝜕 𝑡) - 𝜕 ⃗ 𝐀 𝜕 𝑡

hence $𝑉' = 𝑉 - 𝜕 𝑓 𝜕 𝑡$ .

Possible gauges

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