stylistic:style_transfer
Electric potential

If a system is static, such that magnetic fields are not time-varying, then by Faraday's law, the electric field is curl-free. In this case, one can define an electric potential, that is, a function φ {\displaystyle \varphi } such that E = − ∇ φ {\displaystyle \mathbf {E} =-\nabla \varphi }. This is analogous to the gravitational potential. The difference between the electric potential at two points in space is called the potential difference (or voltage) between the two points.

In general, however, the electric field cannot be described independently of the magnetic field. Given the magnetic vector potential, A, defined so that B = ∇ × A {\displaystyle \mathbf {B} =\nabla \times \mathbf {A} }, one can still define an electric potential φ {\displaystyle \varphi } such that: E = − ∇ φ − ∂ A ∂ t, {\displaystyle \mathbf {E} =-\nabla \varphi -{\frac {\partial \mathbf {A} }{\partial t}},} where ∇ φ {\displaystyle \nabla \varphi } is the gradient of the electric potential and ∂ A ∂ t {\displaystyle {\frac {\partial \mathbf {A} }{\partial t}}} is the partial derivative of A with respect to time.

Faraday's law of induction can be recovered by taking the curl of that equation ∇ × E = − ∂ (∇ × A) ∂ t = − ∂ B ∂ t, {\displaystyle \nabla \times \mathbf {E} =-{\frac {\partial (\nabla \times \mathbf {A})}{\partial t}}=-{\frac {\partial \mathbf {B} }{\partial t}},} which justifies, a posteriori, the previous form for E.