The Weibel instability is a plasma instability present in homogeneous or nearly homogeneous electromagnetic plasmas which possess an anisotropy in momentum (velocity) space. This anisotropy is most generally understood as two temperatures in different directions. Burton Fried showed that this instability can be understood more simply as the superposition of many counter-streaming beams. In this sense, it is like the two-stream instability except that the perturbations are electromagnetic and result in filamentation as opposed to electrostatic perturbations which would result in charge bunching. In the linear limit the instability causes exponential growth of electromagnetic fields in the plasma which help restore momentum space isotropy. In very extreme cases, the Weibel instability is related to one- or two-dimensional stream instabilities. Consider an electron-ion plasma in which the ions are fixed and the electrons are hotter in the y-direction than in x or z-direction. To see how magnetic field perturbation would grow, suppose a field B = B cos kx spontaneously arises from noise. The Lorentz force then bends the electron trajectories with the result that upward-moving-ev x B electrons congregate at B and downward-moving ones at A. The resulting current sheets generate magnetic field that enhances the original field and thus perturbation grows. Weibel instability is also common in astrophysical plasmas, such as collisionless shock formation in supernova remnants and -ray bursts. A Simple Example of Weibel Instability As a simple example of Weibel instability, consider an electron beam with density and initial velocity propagating in a plasma of density with velocity . The analysis below will show how an electromagnetic perturbation in the form of a plane wave gives rise to a Weibel instability in this simple anisotropic plasma system. We assume a non-relativistic plasma for simplicity. We assume there is no background electric or magnetic field i.e. . The perturbation will be taken as an electromagnetic wave propagating along i.e. . Assume the electric field has the form With the assumed spatial and time dependence, we may use and . From Faraday's Law, we may obtain the perturbation magnetic field Consider the electron beam. We assume small perturbations, and so linearize the velocity and density . The goal is to find the perturbation electron beam current density where second-order terms have been neglected. To do that, we start with the fluid momentum equation for the electron beam which can be simplified by noting that and neglecting second-order terms. With