Question

Select One Option Correct from the following : A dense collection of equal number of electrons and positive ions is called neutral plasma. Certain solids containing fixed positive ions sur- rounded by free electrons can be treated as neutral plasma. Let N be number density of free electrons, each of mass m. When the electrons are subjected to an electric field, they are displaced relatively away from the heavy positive ions. If the electric field become zero, the electrons begin to oscillate about positive ions with natural frequency $\omega_p$, which is called plasma frequency. To sustain the oscillations, a time varying electric field needs to be applied that has an angular frequency$\omega$, where a part of energy is absorbed and a part of it is reflected. As$\omega$ approaches $\omega_p$, all the free electrons are set to resonate together and all the energy is reflected. This is the explanation for high reflectivity of metals.Taking the electronic charge as e and permittivity as $\epsilon_0$, use dimen- sional analysis to determine correct expression for $\omega_p$.
(A) $\sqrt{\frac{Ne}{m \epsilon_0} }$
(B) $\sqrt{\frac{m \epsilon_0}{Ne} }$
(C) $\sqrt{\frac{Ne^2}{m \epsilon_0} }$
(D) $\sqrt \frac{m \epsilon_0}{Ne^2}$

Answer 1

answer : option (c)

explanation : N is number of electrons per unit volume,

so, dimension of N = $[L^{-3}]$

e is charge on electron,

so, dimension of e = $[AT]$

m is mass of electron,

so, dimension of m = $[M]$

$\epsilon_0$ is permittivity of medium.

so, dimension of $\epsilon_0$is found by using Coulomb's force.

e.g., F = $\frac{1}{4\pi\epsilon_0}\frac{q_1q_2}{r^2}$

so, dimension of $\epsilon_0$ = dimension of $\frac{q_1q_2}{Fr^2}$

= $\frac{[AT]^2}{[MLT^{-2}][L^2]}$

= $\frac{[A^2T^2]}{[ML^3T^{-2}]}$

= $[M^{-1}L^{-3}T^4A^2]$

and $\omega_p$ is angular frequency.

so, dimension of $\omega_p$ = $[T^{-1}]$

now using dimensional analysis,

$\omega_p=kN^xe^ym^z\epsilon_0^w$

$or,[T^{-1}=k[L^{-3}]^x[AT]^y[M]^z[M^{-1}L^{-3}T^4A^2]^w$

$or,[M^0L^0T^{-1}A^0]=k[M]^{(z-w)}[L]^{(-3x-3w)}[T]^{(y+4w)}[A]^{(y+2w)}$

comparing both sides,

z - w = 0 => z = w .......(1)

-3x - 3w = 0 => x = -w ......(2)

y + 4w = -1 .......(3)

y + 2w = 0 ......(4)

from equation (3) and (4),

2w = -1 , w = -1/2

and y = -2w = 1

from equation (1),

z = w = -1/2

from equation,

x = -w = 1/2

now, $\omega_p=kN^{1/2}e^{1}m^{-1/2}\epsilon_0^{-1/2}$

or, $\omega_p=k\sqrt{\frac{Ne^2}{m\epsilon_0}}$

hence, option (c) is correct.