Solve the following problemsQ.1 Derive an expression for plasma oscillation frequency for ions.Q.2 For a simple plasma oscillation with fixed ions and a space-time behaviorexpli(kx – wt)], calculate the phase ô for Ø1, E1, and v, if the phase of n, iszero.Illustrate the relative phases by drawing sine waves representing n1, Ø1, E1 andv, (a) as a function of x at t = 0, (b) as a function of t at x = 0 for w/k > 0,and (c) as a function of I at x = 0 for w/k < 0. Note that the time patterns canbe obtained by translating the x patterns in the proper direction, as if the wavewere passing by a fixed observer.Q.3. By writing the linearized Poisson's equation used in the derivation of simpleplasma oscillations in the formV . (E E) = 0derive an expression for the dielectric constant E applicable to high frequencylongitudinal motions.

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