= Solution
In SI units, with microscopic <charge density> $\rho_e$ and <current density> $\mathbf J$, <Maxwell's equations> can be written
$$
\boxed{\nabla\cdot\mathbf E=\frac{\rho_e}{\varepsilon_0},\quad\nabla\cdot\mathbf B=0,\quad\nabla\times\mathbf E=-\frac{\partial\mathbf B}{\partial t},\quad\nabla\times\mathbf B=\mu_0\mathbf J+\mu_0\varepsilon_0\frac{\partial\mathbf E}{\partial t}.}
$$
The first relates <electric flux> to charge. The second states that the net <magnetic flux> through a closed surface is zero. <Faraday's law> relates electric <circulation> to changing <magnetic flux>. The <Ampère-Maxwell equation> includes both conduction current and <displacement current>; the latter is essential to time-dependent fields and local <charge conservation>.
For fixed contours and spanning surfaces, the <integral> <curl> equations are $\oint\mathbf E\cdot d\mathbf l=-d\Phi_B/dt$ and $\oint\mathbf B\cdot d\mathbf l=\mu_0I_{\rm enc}+\mu_0\varepsilon_0d\Phi_E/dt$. For a moving material contour the first must include the motional <force> term; its total emf is $\oint(\mathbf E+\mathbf v_{\rm loop}\times\mathbf B)\cdot d\mathbf l$. This prevents confusing a changing circuit <area> with a partial time <derivative> of a static <magnetic field>.
Taking the <divergence> of the <Ampère-Maxwell equation> and using electric <Gauss's law> gives $\partial_t\rho_e+\nabla\cdot\mathbf J=0$. In a source-free <vacuum>, combining the two <curl> equations gives $\nabla^2\mathbf E-\mu_0\varepsilon_0\partial_t^2\mathbf E=0$, and similarly for $\mathbf B$, predicting <electromagnetic waves> of <speed> $c=1/\sqrt{\mu_0\varepsilon_0}$. The <Lorentz force> law then specifies how these fields act on a charge. Macroscopic matter descriptions may instead use $\mathbf D$ and $\mathbf H$ with free sources and constitutive relations; those should not be silently mixed with the microscopic-source equations above.
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