Generalized Ohm’s Law for a Background Plasma in the Presence of Relativistic Charged Particles

A generalized Ohm’s law is derived for a system composed of a background magnetohydrodynamic plasma and a lower density relativistic charged-particle distribution. The interpretation of Ohmic electric fields occurring due to force balance breaks down for such a system and instead an approach based on Maxwell’s equations along with the particle flux equations is necessary. Three additional terms arise in Ohm’s law and each is verified numerically.

Figure 1

The electric field as found in simulations which solve Maxwell’s equations and the particle equations of motion. The fast particles undergo an initial acceleration period up to a relativistic $\gamma =1.5$, shortly after which a magnetic field is applied which induces a harmonic electric field. Also shown are the various contributions to the field from Ohm’s law labeled according to their order of appearance on the rhs of Eq. (12).

Figure 2

The relative magnitude of the two terms in Eq. (14), equivalent to the ratio ${\tau }_{\mathrm{ci}}/({\tau }_0/4)$ as a function of background electron temperature, for a range of densities (normalized to the nonrelativistic critical density of $1\mu \mathrm{m}$ light).

Figure 3

The longitudinal electric field set up as a result of fast-electron pulses streaming from left to right, inside a solid-Al plasma. Also shown is the field given by the Ohm’s law Eq. (14) and its contributions from the first (current-flux tensor) and second (collisional) terms. The data are shown at $t=30\mathrm{fs}$. Note that the spatial range begins inside the dense target.