Question

A student determines a dimensionless quantity,$B=\frac{e^{n}}{2\varepsilon_{0}hc}$ Find the value of $n$ (here, $e =$ electric charge, $\varepsilon_{0}$ electric permittivity of vacuum, $h =$ Planck’s constant and $c =$ speed of light.)

• A. $3$
• B. $4$
• C. $2$
• D. $1$

Answer 1

Answer: (C)

Dimension of $B=\left[M^{0}L^{0}T^{0}\right]$
Dimensions $e=\left[AT\right]$
Dimensions $\varepsilon_{0}=\left[A^{2}M^{-1}L^{-3}T^{4}\right]$
Dimensions $h=\left[ML^{2}T^{-1}\right]$
Dimension $c=\left[LT^{-1}\right]$
$\therefore \left[M^{0}L^{0}T^{0}\right]=\frac{\left[AT\right]^{n}}{\left[A^{2}M^{-1}L^{-3}T^{4}\right]\left[ML^{2}T^{-1}\right]\left[LT^{-1}\right]}$
or $\left[M^{0}L^{0}T^{0}\right]=\left[A^{n-2}L^{0}T^{n-2}\right]$
$\therefore n-2=0$
$\Rightarrow n=2$