Answer: in a medium with charge density \(\rho\) and current density \(\vec J\), Maxwell's equations in differential form are \(\nabla\cdot\vec D = \rho\), \(\nabla\cdot\vec B = 0\), \(\nabla\times\vec E = -\partial\vec B/\partial t\) and \(\nabla\times\vec H = \vec J + \partial\vec D/\partial t\).
- Gauss's law for electricity: \[\nabla\cdot\vec D = \rho\]
- Gauss's law for magnetism: \[\nabla\cdot\vec B = 0\]
- Faraday's law: \[\nabla\times\vec E = -\frac{\partial\vec B}{\partial t}\]
- Ampère–Maxwell law: \[\nabla\times\vec H = \vec J + \frac{\partial\vec D}{\partial t}\]
For a linear, isotropic medium the fields are related by the constitutive relations
\[\vec D = \varepsilon\vec E,\quad \vec B = \mu\vec H,\quad \vec J = \sigma\vec E\]
where \(\varepsilon = \varepsilon_0\varepsilon_r\) is the permittivity, \(\mu = \mu_0\mu_r\) the permeability and \(\sigma\) the conductivity of the medium. The term \(\partial\vec D/\partial t\) is the displacement current density. In free space (\(\rho = 0\), \(\vec J = 0\), \(\varepsilon = \varepsilon_0\), \(\mu = \mu_0\)) these reduce to the source-free equations that give electromagnetic waves travelling at \(c\).