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2.2: Differential Laws of Electrodynamics

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    In the Chu formulation,\(1\) with material effects on the fields accounted for by the magnetization density \(\overrightarrow{M}\) and the polarization density \(\overrightarrow{P}\) and with the material velocity denoted by \(\overrightarrow{v}\), the laws of electrodynamics are:

    Faraday's law

    \[ \nabla \times \overrightarrow{E} = -\mu_o \frac{\partial\overrightarrow{H}}{\partial t} - \mu_o \frac{\partial\overrightarrow{M}}{\partial t} - \mu_o\nabla \times (\overrightarrow{M} \times \overrightarrow{v}) \label{1} \]

    Ampere's law

    \[ \nabla \times \overrightarrow{H} = \varepsilon_o \frac{\partial\overrightarrow{E}}{\partial t} + \frac{\partial\overrightarrow{P}}{\partial t} + \nabla \times (\overrightarrow{P} \times \overrightarrow{v}) + \overrightarrow{J_f} \label{2} \]

    Gauss' law

    \[ \varepsilon_o \nabla \cdot \overrightarrow{E} = - \nabla \cdot \overrightarrow{P} + \rho_f \label{3} \]

    Divergence law for magnetic fields

    \[ \mu_o \nabla \cdot \overrightarrow{H} = - \mu_o\nabla \cdot \overrightarrow{M}\label{4} \]

    and conservation of free charge

    \[ \nabla \cdot \overrightarrow{J_f} + \frac{\partial \rho_f}{\partial t} = 0 \label{5} \]

    This last expression is embedded in Ampere's and Gauss' laws, as can be seen by taking the divergence of Equation \ref{2} and exploiting Equation \ref{3}. In this formulation the electric displacement and magnetic flux density \(\overrightarrow{B}\) are defined fields:

    \[ \overrightarrow{D} = \varepsilon_o \overrightarrow{E} + \overrightarrow{P} \label{6} \]

    \[ \overrightarrow{B} = \mu_o (\overrightarrow{H} + \overrightarrow{M}) \label{7} \]

    This page titled 2.2: Differential Laws of Electrodynamics is shared under a CC BY-NC-SA 4.0 license and was authored, remixed, and/or curated by James R. Melcher (MIT OpenCourseWare) via source content that was edited to the style and standards of the LibreTexts platform; a detailed edit history is available upon request.