### Scalar and Vector Potentials
emf.svp

\begin{align} (i)~~ &{\boldsymbol \nabla} \cdot {\bf E} = \frac{\rho}{\varepsilon_0}, &(iii)~~ {\boldsymbol \nabla} \times {\bf E} + \frac{\partial {\bf B}}{\partial t} &= 0, \nonumber \\ (ii)~~ &{\boldsymbol \nabla} \cdot {\bf B} = 0, &(iv)~~ {\boldsymbol \nabla} \times {\bf B} - \mu_0 \varepsilon_0 \frac{\partial {\bf E}}{\partial t} &= \mu_0 {\boldsymbol J}. \end{align}

Solving these for generale time-dependent sources \(\rho({\boldsymbol r}, t)\) and \({\boldsymbol J} ({\boldsymbol r}, t)\) is not an easy task.

Useful strategy: represent fields in terms of potentials.

Easiest: as we already saw (BcurlA), we can write the magnetic field as a pure curl (since its divergence always vanishes):

\[ {\boldsymbol B} = {\boldsymbol \nabla} \times {\boldsymbol A} \]

Putting this into Faraday's law Fl gives \[ {\boldsymbol \nabla} \times \left({\boldsymbol E} + \frac{\partial {\boldsymbol A}}{\partial t} \right) = 0 \] so this can be written as the gradient of a scalar. Making the choice \(-{\boldsymbol \nabla} \phi\) for this, we get

\[ {\boldsymbol E} = -{\boldsymbol \nabla} \phi - \frac{\partial {\boldsymbol A}}{\partial t} \tag{E_phiA}\label{E_phiA} \]

Using this potential representation for \({\boldsymbol E}\) and \({\boldsymbol B}\) automatically fulfills the two homogeneous Maxwell equations. For the inhomogeneous equations, substituting E_phiA into Gauss's law gives

\[ {\boldsymbol \nabla}^2 \phi + \frac{\partial}{\partial t} {\boldsymbol \nabla} \cdot {\boldsymbol A} = -\frac{\rho}{\varepsilon_0} \tag{Lapphi}\label{Lapphi} \]

whereas Ampère-Maxwell becomes \[ {\boldsymbol \nabla} \times \left({\boldsymbol \nabla} \times {\boldsymbol A}\right) = \mu_0 {\boldsymbol J} - \mu_0 \varepsilon_0 {\boldsymbol \nabla} \left(\frac{\partial \phi}{\partial t}\right) - \mu_0 \varepsilon_0 \frac{\partial^2 {\boldsymbol A}}{\partial t^2} \] which becomes after simple rearrangement and use of the curlcurl identity \({\boldsymbol \nabla} \times \left({\boldsymbol \nabla} \times {\boldsymbol A}\right) = {\boldsymbol \nabla} ({\boldsymbol \nabla} \cdot {\boldsymbol A}) - {\boldsymbol \nabla}^2 {\boldsymbol A}\),

\[ \left( {\boldsymbol \nabla}^2 {\boldsymbol A} - \mu_0 \varepsilon_0 \frac{\partial^2 {\boldsymbol A}}{\partial t^2} \right) - {\boldsymbol \nabla} \left({\boldsymbol \nabla} \cdot {\boldsymbol A} + \mu_0 \varepsilon_0 \frac{\partial \phi}{\partial t} \right) = -\mu_0 {\boldsymbol J} \tag{LapA}\label{LapA} \]

Except where otherwise noted, all content is licensed under a Creative Commons Attribution 4.0 International License.

Author: Jean-Sébastien Caux

Created: 2022-03-24 Thu 08:42