Maxwell equations
= Maxwell equations
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= Maxwell's equations
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The <Maxwell equations> are the field equations of classical <electromagnetism>. In <SI> units, in vacuum with prescribed <charge density> $\rho$ and <current density> $\mathbf J$, they are
$$
\begin{aligned}
\nabla\cdot\mathbf E&=\rho/\epsilon_0,
&\nabla\cdot\mathbf B&=0,\\
\nabla\times\mathbf E&=-\frac{\partial\mathbf B}{\partial t},
&\nabla\times\mathbf B&=\mu_0\mathbf J+\mu_0\epsilon_0\frac{\partial\mathbf E}{\partial t}.
\end{aligned}
$$
Here $\mathbf E$ is the <electric field> in volts per metre, $\mathbf B$ the <magnetic field> in teslas, $\rho$ in coulombs per cubic metre, and $\mathbf J$ in amperes per square metre. The constants are <vacuum permittivity> $\epsilon_0$ and the vacuum value $\mu_0$ of <magnetic permeability>. “Vacuum” specifies the field laws without a material response; “source-free” additionally means $\rho=0$ and $\mathbf J=0$. Matter described by averaged fields instead requires <Maxwell equations in matter> and <constitutive relations>.
The first equation, <Gauss's law>, makes <electric charge> a source of <electric flux>. <Gauss's law for magnetism> sets the net <magnetic flux> through a closed surface to zero. The two <curl> equations, <Faraday's law> and the <Ampère-Maxwell equation>, couple changes of the <electric field> and <magnetic field>. Appropriate <initial conditions> and <boundary conditions> select a physical solution; the two <divergence> equations constrain the initial fields. https://ocw.mit.edu/courses/8-02-physics-ii-electricity-and-magnetism-spring-2007/2d2f227e6fd26ee25f366759e02f03dc_chapte13em_waves.pdf[MIT 8.02 notes, §13.3: Maxwell's equations]{ref}.