2.2: Differential Laws of Electrodynamics
- Page ID
- 28125
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)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:
\[ \overrightarrow{D} = \varepsilon_o \overrightarrow{E} + \overrightarrow{P} \label{6} \]
\[ \overrightarrow{B} = \mu_o (\overrightarrow{H} + \overrightarrow{M}) \label{7} \]