Part 1 : Nanoparticles Module 1 : Introduction
Lecture 1 : Maxwell's equations and Time-harmonic fields
 
The equations above are not sufficient for a given situation/material and must be supplemented with the so called Constitutive Relations, which are assumed to have the form
\begin{displaymath}{\vec{j}}_{\rm free}= {\sigma{\hspace{-0.25 cm}^{\leftrightar... ...}={\varepsilon{\hspace{-0.25 cm}^{\leftrightarrow}}}:{\vec{E}},\end{displaymath}
where ${\sigma{\hspace{-0.25 cm}^{\leftrightarrow}}}\equiv$ conductivity tensor, ${\mu{\hspace{-0.25 cm}^{\leftrightarrow}}}\equiv$ permeability tensor, ${\chi{\hspace{-0.25 cm}^{\leftrightarrow}}} \equiv$ elctric susceptibility tensor, and ${\varepsilon{\hspace{-0.25 cm}^{\leftrightarrow}}}=\epsilon_o(1+{\chi{\hspace{-0.25 cm}^{\leftrightarrow}}})\equiv$ permittivity tensor, which all depend on the material medium. If these are independent of the field, then the medium is said to be linear ; if independent of position (to be more precise, choice of origin for position coordinates), e.g., in a uniform system of infinite extension, the medium is said to be homogeneous ; if independent of direction (e.g., in a uniform system of spherical shape), then the medium is isotropic.
Note that, in general, the symbol `$:$' for example, in ${\vec{B}}= {\mu{\hspace{-0.25 cm}^{\leftrightarrow}}}:{\vec{H}}$ implies the following:
\begin{displaymath}B_i ({\vec x},t)= \sum_j\,\int\,d^3 x'\int_{-\infty}^t dt' \,\mu _{ij} (\vec x,\vec x', t-t') \, H_j ({\vec x'},t'),\end{displaymath}
and so on. Note that above uses the so called causality condition (i.e., physical effects can only follow the cause, independently of the time origin) which requires that $\mu _{ij} ({\vec x},{\vec x'};t,t')\equiv\mu _{ij}(\vec x,\vec x', t-t')$, for $t>t'$, and zero otherwise; for a homogeneous system, one further has $\mu _{ij}(\vec x,\vec x', t-t')\equiv\mu _{ij}(\vec x-\vec x', t-t')$, and for isotropic system, $\mu _{ij} (\vec x,\vec x', t-t')\equiv \mu _{ij} (\vert\vec x-\vec x'\vert, t-t')$; similar arguments apply to ${\sigma{\hspace{-0.25 cm}^{\leftrightarrow}}}$ and ${\varepsilon{\hspace{-0.25 cm}^{\leftrightarrow}}}$, and will be understood in the following developments. Note also that only for a homogeneous medium, for which ${\mu{\hspace{-0.25 cm}^{\leftrightarrow}}}$ is often approximated to be local (its validity is justified only through detailed response theory), i.e., $\mu _{ij} (\vert\vec x-\vec x'\vert, t-t')\equiv \mu _{ij} (t-t')\,\delta(\vec x-\vec x')$, leads to
\begin{displaymath}B_i ({\vec x},t)= \sum_j\,\int_{-\infty}^t dt'\mu _{ij} (t-t') \, H_j ({\vec x},t),\end{displaymath}
and similar relations for other fields.