A dence collection of equal number of electrona and positive ions is called netural plasma. Certain solids contianing fixed positive ions surroundedby free electrons can be treated as neytral plasma. Let 'N' be the numbrer density of free electrons, each of mass '`m`'. When the elctrons are subjected to an eletric field, they are displaced relatively away from the heavy positive ions. if the electric field becomes zero, the electrons begin to oscillate about the positive ions with a natural angular frequency '`omega_(P)'` which is called the plasma frequency. to sustain the oscillations, a time varying electric field needs to be applied that has an angular frequrncy `omega`, where a part of the energy is absorbed and a part of it is reflected. As `omega` approaches `omega_(p)` all the free electrons are set to resonance together and all the energy is reflected. this is the explaination of high reflectivity of metals.
(1) Taking the electronic charge as 'e' and the permittivity as `'epsilon_(0)`'. use dimensional analysis to determine the correct expression for `omega_(p)`.

Options

A. `sqrt((Ne)/(m I"n"_0))`
B. `sqrt((m I"n"_0)/(Ne))`
C. `sqrt((Ne^2)/(m "in"_0))`
D. `sqrt((m I"n"_0)/(Ne^2))`

Answer

(c )

Explanation

Dimensional of , `omega_p = [T^(-1)]`
As `F =(1)/( 4 pi "in"_0) (q^2)/(r^2) or "in"_0 = (q^2)/(4pi r^2 F)`
`sqrt((Ne^2)/( m "in"_0)) = sqrt((Ne^2)/(m)) (4pi r^2 F)/(q^2) = sqrt((N4 pi r^2 F)/(m))`
:. Dimensions of `sqrt((Ne^2)/(m i"n"_0)) = sqrt(([L^(-3)][L^2][MLT^(-2)])/(M))`
`=sqrt((T^(-2))) = [T^(-1)]`
Thus, option (c ) is dimensionally correct.