Question

Taking the electronic charge as e and the permittivity as ${\epsilon }_0,$
use dimensional analysis to determine the correct expression
for ${\omega }_p.$

• B. $\sqrt{\frac{m{\epsilon }_0}{Ne}}$
  
• C. $\sqrt{\frac{N{e}^2}{m{E}_0}}$
  
• D. $\sqrt{\frac{m{\epsilon }_0}{N{e}^2}}$
  

Answer 1

Answer: (C)

N = Number of electrons per unit volume
$\therefore [N]=[L^{-3}],[e]=[q]=[It]=[AT]$
$[{\epsilon }_0]=[M^{-1}{L}^{-3}{T}^4{A}^2]$
Substituting the dimensions we can see that,
$\left[\sqrt{\frac{N{e}^2}{m{\epsilon }_0}}\right]=[T^{-1}]$
Angular frequency has also the dimension $[T^{-1}]$
$\therefore$ Correct option is (c).