Question

A quantity $f$ is given by $f=\sqrt{\frac{h{c}^5}{G}}$ where $c$ is the speed of light, $G$ is the gravitational constant and $h$ is the Planck's constant. The dimension of $f$ is the same as that of:

• A. $\text{area}$
• B. $\text{energy}$
• C. $\text{momentum}$
• D. $\text{volume}$

Answer 1

The correct option is B $\text{energy}$

Dimension of

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

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

$\left[G\right]=\left[M^{-1}{L}^3{T}^{-2}\right]$

Hence, the dimension of $f$ is,

$f=\left[\sqrt{\frac{h{C}^5}{G}}\right]=\sqrt{\frac{\left[M{L}^2{T}^{-1}\right].\left[L^5{T}^{-5}\right]}{\left[M^{-1}{L}^3{T}^{-2}\right]}}$

$=\left[M{L}^2{T}^{-2}\right]=\left[ML{T}^{-2}L\right]$

$=[\text{Force}\times \text{distance}]=\text{energy}$

Hence, $\left(B\right)$ is the correct answer.