Maxwell’s equations unify electricity and magnetism. In vacuum,
\(\varepsilon=\varepsilon_0\) and \(\mu=\mu_0\). In materials (simple linear media),
\(\mathbf{D}=\varepsilon\mathbf{E}\) and \(\mathbf{H}=\mathbf{B}/\mu\).
Constants: \(\mu_0 = 4\pi\times 10^{-7}\ \mathrm{H/m}\), \(\varepsilon_0 \approx 8.854\times 10^{-12}\ \mathrm{F/m}\).
Integral forms
\[
\begin{aligned}
\text{Gauss (E)}\quad &\oint \mathbf{E}\cdot d\mathbf{A} = \frac{Q_{\mathrm{enc}}}{\varepsilon} \\
\text{Gauss (B)}\quad &\oint \mathbf{B}\cdot d\mathbf{A} = 0 \\
\text{Faraday}\quad &\oint \mathbf{E}\cdot d\mathbf{l} = -\frac{d\Phi_B}{dt} \\
\text{Ampère–Maxwell}\quad &\oint \mathbf{B}\cdot d\mathbf{l} = \mu\left(I_{\mathrm{enc}}+\varepsilon\,\frac{d\Phi_E}{dt}\right)
\end{aligned}
\]
Differential forms
\[
\begin{aligned}
\nabla\cdot\mathbf{E} &= \frac{\rho}{\varepsilon} \\
\nabla\cdot\mathbf{B} &= 0 \\
\nabla\times\mathbf{E} &= -\frac{\partial \mathbf{B}}{\partial t} \\
\nabla\times\mathbf{B} &= \mu\left(\mathbf{J}+\varepsilon\,\frac{\partial \mathbf{E}}{\partial t}\right)
\end{aligned}
\]
Why the displacement current matters
In a charging capacitor, current flows in the wires, but there is no conduction current through the insulating gap.
Maxwell’s correction (\(\varepsilon\,d\Phi_E/dt\)) ensures Ampère’s law gives the same magnetic field regardless of which surface you choose.
“Verifier” idea
For symmetric toy models, you can compute both sides of the chosen law and compare them (typically via relative error).
The calculator also shows a representative “local” (differential) check where appropriate.
Website tip: Maxwell’s equations are often presented as the key “unification” step linking E & B into one field theory.
