Perpendicular perturbations in counter-streaming proton beams in cold electron background

Think of a linear, wire-fed antenna and imagine what the fields look like around it. That is basically what a beam is in a plasma but without the mechanical forces keeping the current carrying particles on a specific path.

What you are referring to here is called a two-stream instability. In the simplest scenarios, this will result in parallel propagating (with respect to a quasi-static magnetic field, if present, otherwise along the beams in the zero magnetic field limit), electromagnetic waves. In this simple case, the wave vector is orthogonal to the electric and magnetic field perturbations.

Typically what is done here is to integrate the Vlasov equation over velocity to get the equations of motion of each population. From these you can construct a generalized Ohm's law to relate the particle velocity moments to the electric field.

You can also use these equations of motion to define a current density to then put into Maxwell's equations to solve for a dispersion relation.