Electrodynamics MOC

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=−⃗∇𝑉−𝜕⃗𝐀𝜕𝑡⃗𝐁=⃗∇×⃗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 =⃗∇ ×⃗𝐀. Then inspecting Faraday’s law of induction,

⃗∇×⃗𝐄+𝜕⃗𝐁𝜕𝑡=⃗𝟎⟺⃗∇×⃗𝐄+𝜕𝜕𝑡(⃗∇×⃗𝐀)=⃗𝟎⟺⃗∇×(⃗𝐄+𝜕⃗𝐀𝜕𝑡)=⃗𝟎

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

The electric and magnetic fields are Gauge invariant under the transformation

⃗𝐀→⃗𝐀+⃗∇𝑓𝑉→𝑉−𝜕𝑓𝜕𝑡

Proof

Applying the identity ⃗∇ ×(⃗∇𝑓) =0 it immediately follows that ⃗𝐀′ =⃗𝐀 +⃗∇𝑓 gives the same ⃗𝐁-field as ⃗𝐀, however if we want ⃗𝐄 to also be the same we require that

−⃗∇𝑉−𝜕⃗𝐀𝜕𝑡=−⃗∇𝑉′−𝜕𝜕𝑡(⃗𝐀+⃗∇𝑓)=−∇(𝑉′+𝜕𝑓𝜕𝑡)−𝜕⃗𝐀𝜕𝑡

hence 𝑉′ =𝑉 −𝜕𝑓𝜕𝑡.

Possible gauges

• Coulomb gauge
• Lorenz gauge

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