Path of the Beam of a Relativistic Moving Laser and Some Comments on Modified Coulomb Forces This animation shows the path of a beam from a moving laser that would be observed by fixed observer. The path (trajectory) of the beam can be seen from the z axis as scatter from dust particles in the air. The laser moves along the x axis and can be tilted at any angle, theta, with respect to this axis. The beam trajectory then makes an angle psi with respect to the x axis. The angle psi can be computed as psi=ArcTan[sin(theta)/(cos(theta)-v/c)] where v is the speed of the laser and c is the speed of light. The animation shows photons that are emitted from the laser at angle theta at regular intervals. Prior to its emission, each photon travels with the laser at speed v. After emission the photons travel at speed c in direction theta. I've included a slider that allows the learner to set up a time dependent variation of the emission angle theta. This then results in a curved trajectory Having computed psi, it is possible to also compute the effect of relativity on the forces due to a moving charged particle on a fixed charged particle that is placed off of the x axis at angle theta (which varies with respect to source position) with respect to the moving source. These forces are, of course, the usual magnetic and electric fields except that they have some second order relativistic modifiers. The photons emitted are analogous to entities called "threads" from which we can compute the modified coulomb forces between particles. The equation for these second order force modifiers in terms of psi(theta) is: F=F0(1-beta^2)/[1-(beta*sin(psi))^2]^(3/2) This factor is plotted versus theta as part of the animation. It should be noted that the factor has its greatest value when sin(psi)=1 so that the trajector is perpendicular to the x axis. In addition to the plot of the dependence of force on theta, I have also plotted in red the Force on an off-axis particle versus the x position of the source particle. In this, I assume that both particles are quite massive so neither is displaced significantly during their brief encounter.
beta psiV theta0 dTheta