If the elctron `(` charge of each electron `=-e)` are shifted by a small distance `x,a` net `+ ve` charge density `(` per unit area `)` is induced on the surface. This will result in an electric field `E=n ex //epsilon_(0)` in the direction of `x` and a restoring force on an electron of
`-(n e^(2)x)/(epsilon_(0))`,
Thus `m ddot(x) =-(n e^(2)x)/(epsilon_(0))`
or `ddot(x) + ( n e^(2))/( m epsilon_(0))x=0`
This gives `omega_(p)= sqrt((n e^(2))/( m epsilon_(0)))=1.645xx10^(16)s^(-1)`
as the plasma frequency for the problem,.

Explanation

If the electron (charge of each electron =`-e`) are shifted by a small distance `x`, a net `+ve` charge density (per unit area) is induced on the surface. This will result in an electric field `E=n e x//epsilon_(0)` in the direction of `x` and `a` restoring force on an electron of `-(n e^(2)x)/(epsilon_(0))`,
Thus, `mddot(x)=-(n e^(2)x)/(epsilon_(0))`
or `ddot(x)+(n e^(2))/(m epsilon_(0))x=0`
This gives `omega_(p)=sqrt(n e^(2))/(mepsilon_(0))=1.645xx10^(16) s^(-1)`.
as the plasma frequency for the problem.