PX284 - Q3 - maxwell's equations in matter

the charge density can be separated into two components: free and bound

\begin{matrix} ρ = ρ_{f} + ρ_{b} \\ ρ_{b} = ρ_{P} = - \vec{\nabla} \cdot \vec{P} \end{matrix}

also, the current density has two components:

\begin{matrix} \vec{J} = {\vec{J}}_{f} + {\vec{J}}_{b} \\ {\vec{J}}_{b} = {\vec{J}}_{P} + {\vec{J}}_{M} \\ {\vec{J}}_{P} = \frac{\partial \vec{P}}{\partial t} \\ {\vec{J}}_{M} = \vec{\nabla} \times \vec{M} \end{matrix}

the solenoidal condition remains unchanged:

\vec{\nabla} \cdot \vec{B} = 0

faraday's law does not involve $ρ$ or $\vec{J}$ , so it remains unchanged as well:

\vec{\nabla} \times \vec{E} = - \frac{\partial \vec{B}}{\partial t}

gauss' law:

\begin{aligned} \vec{\nabla} \cdot \vec{E} & = \frac{ρ}{ε_{0}} \\ = \frac{ρ_{f} + ρ_{b}}{ε_{0}} \\ = \frac{ρ_{f} - ρ_{P}}{ε_{0}} \\ = \frac{ρ_{f} - \vec{\nabla} \cdot \vec{P}}{ε_{0}} \end{aligned}

\begin{matrix} \vec{\nabla} \cdot (ε_{0} \vec{E} + \vec{P}) = ρ_{f} \\ \vec{D} = ε_{0} \vec{E} + \vec{P} \end{matrix}

$\vec{D}$ is called the 'displacement', and gauss' law can be rewritten as:

\vec{\nabla} \cdot \vec{D} = ρ_{f}

ampere-maxwell law:

\begin{aligned} \vec{\nabla} \times \vec{B} & = μ_{0} (\vec{J} + ε_{0} \frac{\partial \vec{E}}{\partial t}) \\ = μ_{0} ({\vec{J}}_{f} + {\vec{J}}_{P} + {\vec{J}}_{M} + ε_{0} \frac{\partial \vec{E}}{\partial t}) \\ = μ_{0} ({\vec{J}}_{f} + \frac{\partial \vec{P}}{\partial t} + \vec{\nabla} \times \vec{M} + ε_{0} \frac{\partial \vec{E}}{\partial t}) \\ ⟹ \vec{\nabla} \times (\frac{\vec{B}}{μ_{0}} - \vec{M}) & = {\vec{J}}_{f} + \frac{\partial}{\partial t} (ε_{0} \vec{E} + \vec{P}) \\ \vec{H} & = \frac{\vec{B}}{μ_{0}} - \vec{M} \end{aligned}

$\vec{H}$ is called the 'magnetic field strength', and $\vec{B}$ is called the 'magnetic flux density'

\vec{\nabla} \times \vec{H} = {\vec{J}}_{f} + \frac{\partial \vec{D}}{\partial t}

the complete set of maxwell's equations in matter:

\begin{matrix} \vec{\nabla} \cdot \vec{D} = ρ_{f} \\ \vec{\nabla} \cdot \vec{B} = 0 \\ \vec{\nabla} \times \vec{E} = - \frac{\partial \vec{B}}{\partial t} \\ \vec{\nabla} \times \vec{H} = {\vec{J}}_{f} + \frac{\partial \vec{D}}{\partial t} \end{matrix}

for the electric case:

\vec{P} = ε_{0} χ \vec{E}

isotropic: $\vec{P} ∥ \vec{E}$
linear: $χ \neq χ (\vec{E})$

\vec{D} = ε_{0} \vec{E} + \vec{P} = ε_{0} (1 + χ) \vec{E} = ε_{0} ε_{r} \vec{E}

for the magnetic case:

\vec{M} = χ_{M} \vec{H}

where, $χ_{M}$ is the magnetic susceptibility

isotropic: $\vec{M} ∥ \vec{H}$
linear: $χ \neq χ (\vec{H})$

\vec{B} = μ_{0} (\vec{H} + \vec{M}) = μ_{0} (1 + χ_{M}) \vec{M} = μ_{0} μ_{r} \vec{M}

$χ$ and $χ_{M}$ may be freqency dependent
eg: water molecules (H $_{2}$ O)
the atoms are aligned at an angle, so there is a dipole

in the absence of an electric field, the dipole moments have random orientations due to thermal motion, giving a net zero dipole moment
when an external field is applied, $\vec{E}$ , the diploes align with the field, giving a net polarization along the direction of $\vec{E}$

if $\vec{E}$ is oscillating, molecules/ions need time to reorient themselves, which is les possible at high $ω$
therefore, $χ = χ (ω)$ and $ε_{r} = ε_{r} (ω)$ , and $χ_{M} = χ_{M} (ω)$ and $μ_{r} = μ_{r} (ω)$