Question

Taking the electronic charge as e and the permittivity as $\varepsilon_0, $
use dimensional analysis to determine the correct expression
for $\omega_p. $

• B. $\sqrt {\frac {m\varepsilon_0}{Ne}} $
• C. $\sqrt {\frac {Ne^2}{mE_0}} $
• D. $\sqrt {\frac {m\varepsilon_0}{Ne^2}}$

Answer 1

Answer: (C)

N = Number of electrons per unit volume
$\therefore \, \, \, \, \, \, [N]=[L^{-3}],[e]=[q]=[It]=[AT] $
$\, \, \, \, \, \, \, [\varepsilon_0]=[M^{-1}L^{-3}T^4A^2]$
Substituting the dimensions we can see that,
$\, \, \, \, \, \, \, \bigg [\sqrt {\frac {Ne^2}{m\varepsilon_0}}\bigg ]=[T^{-1}] $
Angular frequency has also the dimension $[T^{-1}] $
$\therefore $ Correct option is (c).