stylistic:tone, linguistic:translation
Gauge freedom

The electrostatic potential could have any constant added to it without affecting the electric field. In electrodynamics, the electric potential has infinitely many degrees of freedom. For any (possibly time-varying or space-varying) scalar field, 𝜓, we can perform the following gauge transformation to find a new set of potentials that produce exactly the same electric and magnetic fields:

V ′ = V − ∂ ψ ∂ t A ′ = A + ∇ ψ {\displaystyle {\begin{aligned}V^{\prime }&=V-{\frac {\partial \psi }{\partial t}}\\\mathbf {A} ^{\prime }&=\mathbf {A} +\nabla \psi \end{aligned}}}

Given different choices of gauge, the electric potential could have quite different properties. In the Coulomb gauge, the electric potential is given by Poisson's equation

∇ 2 V = − ρ ε 0 {\displaystyle \nabla ^{2}V=-{\frac {\rho }{\varepsilon _{0}}}}

just like in electrostatics. However, in the Lorenz gauge, the electric potential is a retarded potential that propagates at the speed of light and is the solution to an inhomogeneous wave equation:

∇ 2 V − 1 c 2 ∂ 2 V ∂ t 2 = − ρ ε 0 {\displaystyle \nabla ^{2}V-{\frac {1}{c^{2}}}{\frac {\partial ^{2}V}{\partial t^{2}}}=-{\frac {\rho }{\varepsilon _{0}}}}