Editing Electric potential (section)

=== Gauge freedom ===
{{Main articles|Gauge fixing}}
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, {{math|𝜓}}, we can perform the following [[Gauge Transformation|gauge transformation]] to find a new set of potentials that produce exactly the same electric and magnetic fields:<ref>{{Cite book|last=Griffiths|first=David J.|title=Introduction to Electrodynamics|publisher=Prentice Hall|year=1999|isbn=013805326X|edition=3rd|pages=420}}</ref>

<math display="block">\begin{align}
V^\prime &= V - \frac{\partial\psi}{\partial t} \\
\mathbf{A}^\prime &= \mathbf{A} + \nabla\psi
\end{align}</math>

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

<math display="block">\nabla^2 V=-\frac{\rho}{\varepsilon_0} </math>

just like in electrostatics. However, in the [[Lorenz gauge condition|Lorenz gauge]], the electric potential is a [[retarded potential]] that propagates at the speed of light and is the solution to an [[Inhomogeneous electromagnetic wave equation#A and %CF%86 potential fields|inhomogeneous wave equation]]:

<math display="block">\nabla^2 V - \frac{1}{c^2}\frac{\partial^2 V}{\partial t^2} = -\frac{\rho}{\varepsilon_0} </math>