Question

$m,e,{\epsilon }^{0}h\text{and}c$ denote mass electron, charge of electron, Planck's constant and speed of light, respectively. The dimensions of $\frac{{\text{me}}^4}{{\epsilon }_0^2{\text{h}}^3\text{c}}$ are

• A. $\left[M^0{L}^0{T}^{-1}\right]$
• B. $\left[M^0{L}^{-1}{T}^0\right]$
• C. $\left[M^0{L}^{-1}{T}^{-1}\right]$
• D. $\left[{\text{M}}^2{\text{LT}}^{-3}\right]$

Answer 1

The correct option is B $\left[M^0{L}^{-1}{T}^0\right]$

Given, $\left[\frac{m{e}^4}{{\epsilon }_0^2{h}^{3}C}\right]$
$m=\text{Mass of electron}\left[M^1\right]$
$e=\text{charge on electron}\left[IT\right]$
${\epsilon }^0=\text{permittivity of free space}$
$\left[M^{-1}{L}^{-3}{T}^4{I}^{-2}\right]$
$\text{h = plank constant}[M{L}^2{\text{T}}^{-1}$]
$\text{c = speed of light}\left[L{T}^{-1}\right]$
Put the dimension in given equation,
$\frac{\left[M\right][TI{]}^4}{\left[M^{-1}{L}^{-3}{T}^4{L}^2{]}^2\right[M{L}^2{T}^{-1}{]}^3\left[L{T}^{-1}\right]}$
$\frac{\left[M\right]\left[T^4{I}^4\right]}{\left[M^{-2}{L}^{-6}{T}^8{l}^4\right]\left[M^3{L}^6{T}^{-3}\right]\left[L{T}^{-1}\right]}$
$\left[M^0{L}^{-1}{T}^0\right]$