15 May 2025
Maxwell Equations, Displacement Current, and Potentials
First-principles derivation of Maxwell's equations, current continuity, and electromagnetic potentials.
We use SI units and let $\hat{\mathbf n}$ point outward from a closed surface. In a linear, isotropic medium,
\[\mathbf D=\epsilon\mathbf E,\qquad \mathbf B=\mu\mathbf H,\qquad \mathbf J=\sigma\mathbf E.\]Here $\mathbf E$ is in $\mathrm{V\,m^{-1}}$, $\mathbf D$ in $\mathrm{C\,m^{-2}}$, $\mathbf B$ in tesla, $\mathbf H$ in $\mathrm{A\,m^{-1}}$, $\epsilon$ in $\mathrm{F\,m^{-1}}$, $\mu$ in $\mathrm{H\,m^{-1}}$, and $\mathbf J$ in $\mathrm{A\,m^{-2}}$.
Electric and magnetic Gauss laws
For a point charge $q$ in a homogeneous medium, Coulomb’s field is
\[\mathbf E=\frac{q}{4\pi\epsilon r^2}\hat{\mathbf r}.\]Its flux through a sphere is $E(4\pi r^2)=q/\epsilon$. Superposition extends this result to any charge distribution and any closed surface $S$:
\[\boxed{\oint_S\mathbf D\cdot d\mathbf a=Q_{\mathrm f,enc}}.\]With $Q_{\mathrm f,enc}=\int_V\rho_{\mathrm f}\,dV$ and the divergence theorem,
\[\int_V(\nabla\cdot\mathbf D-\rho_{\mathrm f})\,dV=0.\]The volume is arbitrary, so the integrand vanishes:
\[\boxed{\nabla\cdot\mathbf D=\rho_{\mathrm f}}.\]Magnetic field lines have no observed beginning or end. Their flux through every closed surface is therefore zero:
\[\boxed{\oint_S\mathbf B\cdot d\mathbf a=0} \quad\Longleftrightarrow\quad \boxed{\nabla\cdot\mathbf B=0}.\]Faraday law and its sign
For a fixed contour $C$ bounding a fixed surface $S$, Faraday’s induction law is
\[\oint_C\mathbf E\cdot d\boldsymbol\ell =-\frac{d}{dt}\int_S\mathbf B\cdot d\mathbf a.\]The contour direction and surface normal obey the right-hand rule. The minus sign is Lenz’s law: the induced circulation opposes the change of magnetic flux. Moving the derivative inside the fixed surface and applying Stokes’ theorem gives
\[\int_S\left(\nabla\times\mathbf E+\frac{\partial\mathbf B}{\partial t}\right)\cdot d\mathbf a=0.\]Because the surface is arbitrary,
\[\boxed{\nabla\times\mathbf E=-\frac{\partial\mathbf B}{\partial t}}.\]Why Ampere’s law needs displacement current
The magnetostatic equation $\nabla\times\mathbf H=\mathbf J$ cannot describe time-dependent charge. Taking its divergence would give $\nabla\cdot\mathbf J=0$, whereas conservation of charge requires
\[\boxed{\nabla\cdot\mathbf J=-\frac{\partial\rho_{\mathrm f}}{\partial t}}.\]Use $\rho_{\mathrm f}=\nabla\cdot\mathbf D$ in the continuity equation:
\[\nabla\cdot\left(\mathbf J+\frac{\partial\mathbf D}{\partial t}\right)=0.\]The current density that can consistently source a curl is therefore
\[\mathbf J_{\mathrm{total}}=\mathbf J+\mathbf J_d, \qquad \boxed{\mathbf J_d=\frac{\partial\mathbf D}{\partial t}}.\]$\mathbf J_d$ has the same unit $\mathrm{A\,m^{-2}}$ as conduction-current density. The corrected law is
\[\boxed{\nabla\times\mathbf H =\mathbf J+\frac{\partial\mathbf D}{\partial t}},\]or, in integral form,
\[\boxed{\oint_C\mathbf H\cdot d\boldsymbol\ell =\int_S\mathbf J\cdot d\mathbf a +\frac{d}{dt}\int_S\mathbf D\cdot d\mathbf a}.\]For a charging parallel-plate capacitor of plate area $A$, neglecting fringing,
\[D=\frac{Q}{A},\qquad I_d=\frac{d}{dt}\int_A\mathbf D\cdot d\mathbf a =\frac{d}{dt}(DA)=\frac{dQ}{dt}=I.\]Thus a surface crossing the wire and a surface bulging through the capacitor gap give the same magnetic circulation.
Maxwell equations together
The four macroscopic equations, with free charge and free conduction current as sources, are
\[\boxed{\begin{aligned} \nabla\cdot\mathbf D&=\rho_{\mathrm f}, &\qquad \nabla\cdot\mathbf B&=0,\\ \nabla\times\mathbf E&=-\frac{\partial\mathbf B}{\partial t}, &\nabla\times\mathbf H&=\mathbf J+\frac{\partial\mathbf D}{\partial t}. \end{aligned}}\]Their integral forms are
\[\boxed{\begin{aligned} \oint_S\mathbf D\cdot d\mathbf a&=Q_{\mathrm f,enc}, &\oint_S\mathbf B\cdot d\mathbf a&=0,\\ \oint_C\mathbf E\cdot d\boldsymbol\ell&=-\frac{d}{dt}\int_S\mathbf B\cdot d\mathbf a, &\oint_C\mathbf H\cdot d\boldsymbol\ell&=I_{\mathrm f,enc}+\frac{d}{dt}\int_S\mathbf D\cdot d\mathbf a. \end{aligned}}\]Taking the divergence of the Ampere-Maxwell equation immediately reproduces charge continuity; the correction is therefore required, not optional.
Scalar and vector potentials
Because $\nabla\cdot\mathbf B=0$, a vector potential $\mathbf A$ exists locally such that
\[\boxed{\mathbf B=\nabla\times\mathbf A}.\]Substitution in Faraday’s law gives
\[\nabla\times\left(\mathbf E+\frac{\partial\mathbf A}{\partial t}\right)=0.\]A curl-free field is a gradient. Defining the scalar potential $\phi$ with the electrostatic sign convention,
\[\boxed{\mathbf E=-\nabla\phi-\frac{\partial\mathbf A}{\partial t}}.\]$\phi$ is measured in volts and $\mathbf A$ in $\mathrm{Wb\,m^{-1}}=\mathrm{V\,s\,m^{-1}}$. The same fields result from
\[\boxed{\mathbf A'=\mathbf A+\nabla\chi, \qquad \phi'=\phi-\frac{\partial\chi}{\partial t}},\]because $\nabla\times\nabla\chi=0$ and the added terms in $\mathbf E$ cancel. This is gauge freedom.
In a homogeneous medium choose the Lorenz gauge
\[\boxed{\nabla\cdot\mathbf A+\mu\epsilon\frac{\partial\phi}{\partial t}=0}.\]Gauss’s law then gives
\[-\nabla^2\phi-\frac{\partial}{\partial t}(\nabla\cdot\mathbf A) =\frac{\rho_{\mathrm f}}{\epsilon},\]and hence
\[\boxed{\left(\nabla^2-\mu\epsilon\frac{\partial^2}{\partial t^2}\right)\phi =-\frac{\rho_{\mathrm f}}{\epsilon}}.\]Similarly, substituting $\mathbf B=\nabla\times\mathbf A$ and $\mathbf E=-\nabla\phi-\partial_t\mathbf A$ in the Ampere-Maxwell law, then using the Lorenz gauge, gives
\[\boxed{\left(\nabla^2-\mu\epsilon\frac{\partial^2}{\partial t^2}\right)\mathbf A =-\mu\mathbf J}.\]In source-free regions both potentials propagate with speed $1/\sqrt{\mu\epsilon}$.
Discussion