Question

Planck’s constant (h), speed of light in vacuum (c) and Newton’s gravitational constant (G) are three fundamental constants. Which one of the following combinations of these has the dimension of length?

• A. $\sqrt{\frac{hG}{C^3}}$
• B. $\sqrt{\frac{hG}{C^{5/2}}}$
• C. $\sqrt{\frac{hC}{G}}$
• D. $\sqrt{\frac{GC}{h^{3/2}}}$

Answer 1

The correct option is A $\sqrt{\frac{hG}{C^3}}$

$h=\lambda \times mv$

$\left[h\right]=[L\times M\times \frac{L}{T}]=\left[\frac{M{L}^2}{T}\right]$ --- (I)

$G=\frac{F\times r^2}{m_1{m}_2}$
$\left[G\right]=\frac{\left[ML{T}^{-2}\right]\times \left[L^2\right]}{\left[M^2\right]}=\left[\frac{L^3}{M{T}^2}\right]$ --- (II)

$\left[C\right]=\left[\frac{L}{T}\right]$ ---(III)

$\left[hG\right]=\left[\frac{M{L}^2}{T}\right]\left[\frac{L^3}{M{T}^2}\right]=\left[\frac{L^5}{T^3}\right]=\left[L^2\right]\times \left[\frac{L^3}{T^3}\right]$

i.e $\left[hG\right]=\left[L^2\right]\times \left[C^3\right]$

i.e $\left[\sqrt{\frac{hG}{C^3}}\right]=\left[L\right]$