When the macroscopic neutrality of a plasma is instantaneously perturbed by external means, the electrons react in a such a way as to give rise to oscillations at the electron plasma frequency wpe=(n092/meco)1/2. consider these oscillations, but including the motion of the ions. show that in this case the natural frequency of oscillation of the net charge density is given by w = (we + %)/2 where wpi=(no92/m.€o)/2. use the linearized equations of continuity and of momen- tum for each species, and poisson equation, considering only the electric force due to the internal charge separation.

As you are trying to move a heavy box of mass m, you realize that it is too heavy for you to lift by yourself. there is no one around to , so you attach an ideal pulley to the box and a massless rope to the ceiling, which you wrap around the pulley. you pull up on the rope to lift the box. use g for the magnitude of the acceleration due to gravity and neglect friction forces. once you have pulled hard enough to start the box moving upward, what is the magnitude f of the upward force you must apply to the rope to start raising the box with constant velocity? express the magnitude of the force in terms of m, the mass of the box.