The magnetic flux is Φ M = ∫ B ⃗ ⋅ d A ⃗ Φ M = ∫ B ⃗ ⋅ d A ⃗ A device that can maintain a potential difference, despite the flow of current is a source of electromotive force. (EMF) The definition is mathematically ɛ = ∮ E ⃗ ⋅ d s ⃗ ɛ = ∮ E ⃗ ⋅ d s ⃗ Faraday's law states that ɛ = − ∂ Φ M ∂ t ɛ = − ∂ Φ M ∂ t This is simply an experimental fact.

Faraday's law could also be written ∮ E ⃗ ⋅ d s ⃗ = − ∂ ∂ t ∫ B ⃗ ⋅ d A ⃗ ∮ E ⃗ ⋅ d s ⃗ = − ∂ ∂ t ∫ B ⃗ ⋅ d A ⃗

I will leave it as a problem to show that this can also be written ∇ ⃗ × E ⃗ = − ∂ B ⃗ ∂ t . ∇ ⃗ × E ⃗ = − ∂ B ⃗ ∂ t .