Wake excited in plasma by an ultrarelativistic pointlike bunch

We study propagation of a relativistic electron bunch through a cold plasma assuming that the transverse and longitudinal dimensions of the bunch are much smaller than the plasma collisionless skin depth. Treating the bunch as a point charge and assuming that its charge is small, we derive a simplified system of equations for the plasma electrons and show that, through a simple rescaling of variables, the bunch charge can be eliminated from the equations. The equations demonstrate an ion cavity formed behind the driver. They are solved numerically and the scaling of the cavity parameters with the driver charge is obtained. A numerical solution for the case of a positively charged driver is also found.