Maxwell’s equations
The standard Maxwell equations
Charge densities \(\require{physics} \rho(\vb{r}, t)\), current densities \(\vb{j}(\vb{r}, t)\), electric fields \(\vb{E}(\vb{r}, t)\) and magnetic fields \(\vb{B}(\vb{r}, t)\) are fundamentally connected in nature. Mathematically, we can describe that connection by Maxwell’s equations (in SI-based atomic units) (T. Helgaker et al. 2012): \[\begin{align} & \boldsymbol{\nabla} \vdot{\vb{B}(\vb{r}, t)} = 0 \\ % & \boldsymbol{\nabla} \vdot{\vb{E}(\vb{r}, t)} = \frac{\rho(\vb{r}, t)}{\epsilon_0} \\ % & \curl{\vb{E}(\vb{r}, t)} = - \pdv{\vb{B}(\vb{r}, t)}{t} \\ % & \curl{\vb{B}(\vb{r}, t)} = \mu_0 \thinspace \vb{j}(\vb{r}, t) + \frac{1}{c^2} \pdv{\vb{E}(\vb{r}, t)}{t} \thinspace , \end{align}\] which are consistent with the (charge) continuity equation \[\begin{equation} \pdv{\rho(\vb{r}, t)}{t} + \boldsymbol{\nabla} \vdot{\vb{j}(\vb{r}, t)} = 0 \end{equation}\] and Lorentz’ force \[\begin{equation} \vb{F}(\vb{r}, t) = q \qty[ \vb{E}(\vb{r}, t) + \vb{v}(\vb{r}, t) \cross \vb{B}(\vb{r}, t) ] \thinspace . \end{equation}\] The vacuum permittivity \(\epsilon_0\) and the vacuum permeability \(\mu_0\) are related by the speed of light \(c\): \[\begin{equation} \epsilon_0 \mu_0 = \frac{1}{c^2} \thinspace . \end{equation}\] The first and third equations and are called the homogeneous Maxwell equations and the second and fourth are then called the inhomogeneous Maxwell equations because the differential equations of \(\vb{E}(\vb{r}, t)\) and \(\vb{B}(\vb{r}, t)\) are polluted by the inhomogeneities (sources) \(\rho(\vb{r}, t)\) and \(\vb{j}(\vb{r}, t)\).
As a consequence of the homogeneous Maxwell equations and differential vector calculus, we can write the electric and magnetic fields as \[\begin{align} & \vb{E}(\vb{r}, t) = - \grad{\phi(\vb{r}, t)} - \pdv{\vb{A}(\vb{r}, t)}{t} \\ % & \vb{B}(\vb{r}, t) = \curl{\vb{A}(\vb{r}, t)} \thinspace , \end{align}\] in which \(\phi(\vb{r}, t)\) is called the scalar potential and \(\vb{A}(\vb{r}, t)\) is called the vector potential.For uniform electric or magnetic fields, the associated scalar and vector potentials can be written in a simple form.
The scalar and vector potentials then satisfy the inhomogeneous Maxwell equations, leading to \[\begin{align} & - \laplacian{\phi(\vb{r}, t)} - \pdv{ (\boldsymbol{\nabla} \vdot{\vb{A}(\vb{r}, t)}) }{t} = \frac{\rho(\vb{r}, t)}{\epsilon_0} \\ % & - \laplacian{\vb{A}(\vb{r}, t)} + \frac{1}{c^2} \pdv[2]{ \vb{A}(\vb{r}, t) }{t} + \grad{ \qty[ \frac{1}{c^2} \pdv{ \phi(\vb{r}, t) }{t} + \boldsymbol{\nabla} \vdot{\vb{A}(\vb{r}, t)} ] } = \mu_0 \thinspace \vb{j}(\vb{r}, t) \thinspace . \end{align}\]
Maxwell’s equations for the potentials (Lorenz gauge)
In Lorenz gauge, the inhomogeneous Maxwell equations take the following form: \[\begin{align} & - \laplacian{\phi(\vb{r}, t)} + \frac{1}{c^2} \pdv[2]{ \phi(\vb{r}, t) }{t} = \frac{\rho(\vb{r}, t)}{\epsilon_0} \\ % & - \laplacian{\vb{A}(\vb{r}, t)} + \frac{1}{c^2} \pdv[2]{ \vb{A}(\vb{r}, t) }{t} = \mu_0 \vb{j}(\vb{r}, t) \thinspace , \end{align}\] which is a form that hints at a deeper relationship between the scalar potential and the vector potential.
Maxwell’s equations for the potentials (Coulomb gauge)
In Coulomb gauge, the inhomogeneous Maxwell equations become: \[\begin{align} \laplacian{\phi(\vb{r}, t)} & = - \frac{\rho(\vb{r}, t)}{\epsilon_0} \\ % - \laplacian{\vb{A}(\vb{r}, t)} + \frac{1}{c^2} \pdv[2]{ \vb{A}(\vb{r}, t) }{t} + \qty( \frac{1}{c^2} \pdv{ \grad{ \phi(\vb{r}, t) } }{t} ) & = \mu_0 \vb{j}(\vb{r}, t) \thinspace . \end{align}\] The inhomogeneous Maxwell equation for the scalar potential in Coulomb gauge is also called Poisson’s equation. From this equation, we can determine the scalar potential given a charge distribution as \[\begin{equation} \phi(\vb{r}, t) = \int \dd{\vb{r}'} \frac{\rho(\vb{r}', t)}{ \norm{\vb{r} - \vb{r}'} } \thinspace . \end{equation}\] A point charge with charge \(q\), located at \(\vb{r}_0\), has a charge distribution \[\begin{equation} \rho(\vb{r}, t) = q \delta(\vb{r} - \vb{r}_0) \end{equation}\] and thus it generates a corresponding scalar potential: \[\begin{equation} \phi(\vb{r}, t) = \frac{q}{|\vb{r} - \vb{r}_0|} \thinspace . \end{equation}\]
For time-independent systems, the vector potential also satisfies a Poisson-like equation: \[\begin{equation} \laplacian{\vb{A}(\vb{r})} = - \mu_0 \vb{j}(\vb{r}) \thinspace , \end{equation}\] which has the solution \[\begin{equation} \vb{A}(\vb{r}) = \frac{\mu_0}{4 \pi} \int \dd{\vb{r}'} \frac{ \vb{j}(\vb{r}') }{ \norm{ \vb{r} - \vb{r}' } } \thinspace . \end{equation}\] Taking the curl of both sides, we find Biot-Savart’s law.