The following pages describe how to make charge distributions and current measurements within metals.

Note: This analysis should only be applied to real metals (i.e. materials where it is reasonable to assume the material is a cloud of free electrons). This analysis will not work for PEC materials. |

## Background

To calculate the charge distributions and current densities, we treat each metal as a cloud of free electrons, i.e. a plasma. To calculate the current density in a plasma we first recognize that all material properties within the FDTD simulation are implemented via an effective material permittivity:

$$D = \varepsilon _{material} E$$

For the purposes of this calculation, we assume there are two contributions to the material permittivity: the background permittivity (in this case, the permittivity of free space) plus the contribution from the current density.

$$D = (\varepsilon _0 + \varepsilon _{plasma}) E$$

The first term is Do, the displacement field that would exist in free space for the given electric field. The second term is proportional to the current J. For more details, see the notes below.

\begin{aligned} D & = \varepsilon _{material} E\\ & = \varepsilon _0 E + \varepsilon _{plasma} E\\ & = D_0 + \frac{iJ}{\omega} \end{aligned}

Solving the above equation for J, we get

\begin{aligned} J & = -i \omega (D - D_0)\\ & = -i \omega (\varepsilon _{material} - \varepsilon _0)E \end{aligned}

Note: The total material permittivity is created from two contributions: one from the polarization of the medium due to bound charge, and one from the current density due to free charge. Ampere's law can be written as $$-i \omega D_{background} = \nabla \times H - J$$ $$D_{background} = \varepsilon _0 E + P = \varepsilon _0 (1 + \chi) E$$ We can rewrite this equation as \begin{aligned}-i \omega D _{background} &= \nabla \times H - J \\ -i \omega \left(D_{background} + \frac{iJ}{\omega} \right) &= \nabla \times H \\ -i \omega D_{material} &= \nabla \times H \end{aligned} and, assuming that J is proportional to E, we have \begin{aligned} D_{material} &= \varepsilon _0 E + P + \frac{iJ}{\omega} \\ &=(\varepsilon _0 + \varepsilon _0 \chi + \varepsilon _{plasma}) E \\ &= \varepsilon _{material} E \end{aligned} For plasma materials, such as metals, we have assumed that the susceptibility of the medium is 0 (i.e. P=0). |