the plane wave assumption for the derivatives, the momentum equation becomes We can decompose the above equations in components, paying attention to the cross product at the far right, and obtain the non-zero components of the beam velocity perturbation: To find the perturbation density , we use the fluid continuity equation for the electron beam which can again be simplified by noting that and neglecting second-order terms. The result is Using these results, we may use the equation for the beam perturbation current density given above to find Analogous expressions can be written for the perturbation current density of the left-moving plasma. By noting that the x-component of the perturbation current density is proportional to , we see that with our assumptions for the beam and plasma unperturbed densities and velocities the x-component of the net current density will vanish, whereas the z-components, which are proportional to , will add. The net current density perturbation is therefore The dispersion relation can now be found from Maxwell's Equations: where is the speed of light in free space. By defining the effective plasma frequency , the equation above results in This bi-quadratic equation may be easily solved to give the dispersion relation In the search for instabilities, we look for ( is assumed real). Therefore, we must take the dispersion relation/mode corresponding to the minus sign in the equation above. To gain further insight on the instability, it is useful to harness our non-relativistic assumption to simplify the square root term, by noting that The resulting dispersion relation is then much simpler is purely imaginary. Writing we see that , indeed corresponding to an instability. The electromagnetic fields then have the form Therefore, the electric and magnetic fields are out of phase, and by noting that so we see this is a primarily magnetic perturbation although there is a non-zero electric perturbation. The magnetic field growth results in the characteristic filamentation structure of Weibel instability. Saturation will happen when the growth rate is on the order of the electron cyclotron frequency References Lecture See also Chromo-Weibel Instability Plasma instabilities Particle physics