# Definition The magnetic vector potential in electromagnetism $\pmb{A}(pmb{r})$ is defined such that $ \begin{align} \nabla \times \pmb{A}(\pmb{r}) = \pmb{B}(\pmb{r}), \\ E(\pmb{x}) = -\nabla \phi(\pmb{r}) - \frac{\partial \pmb{A}(\pmb{r})}{t} \end{align} $ where $\pmb{B}(\pmb{r})$ is the [[magnetic field|magnetic field]], and $\pmb{E}(\pmb{r})$ is the [[electric field|electric field]]. $\phi(\pmb{r})$ is the [[electric potential|electric potential]] a [[scalar]] field. More generally, a vector potential is a [[vector field]] whose [[Curl|curl]] is a given [[vector field]]. # Gauge Transformation We will omit the argument $\pmb{r}$ from here on out. The transformation $ \pmb{A} \rightarrow \pmb{A} + \nabla \psi $ for some [[scalar]] field $\psi$ leads to: $ \begin{align} \pmb{B} \rightarrow \pmb{B}, \\ \pmb{E} \rightarrow \pmb{E} - \nabla \frac{\partial \psi}{\partial t}, \\ \end{align} $ Or, we can leave $\pmb{E}$ unchanged if we redefine the scalar potential as: $ \phi \rightarrow \phi - \frac{\partial \psi}{\partial t} $ ## Coulomb Gauge ## Lorenz Gauge