Question

A student determines a dimensionless quantity,$B=\frac{e^n}{2{\epsilon }_{0}hc}$ Find the value of $n$ (here, $e=$ electric charge, ${\epsilon }_0$ 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 ${\epsilon }_0=\left[A^2{M}^{-1}{L}^{-3}{T}^4\right]$

Dimensions $h=\left[M{L}^2{T}^{-1}\right]$

Dimension $c=\left[L{T}^{-1}\right]$

$\therefore \left[M^0{L}^0{T}^0\right]=\frac{{\left[AT\right]}^n}{\left[A^2{M}^{-1}{L}^{-3}{T}^4\right]\left[M{L}^2{T}^{-1}\right]\left[L{T}^{-1}\right]}$

or $\left[M^0{L}^0{T}^0\right]=\left[A^{n-2}{L}^0{T}^{n-2}\right]$

$\therefore n-2=0$

$\Rightarrow n=2$