Physics

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

$\dfrac{\sqrt{hG}}{c^{3/2}}$

SOLUTION
The SI unit of h is $J.s$ or $N.m.s$ and SI unit of c is $m/s$ and Si unit of G is $N.m^2/kg^2$

For A: $\sqrt{\dfrac{(N.m^2/kg^2)(m/s)}{(N.m.s)^{3/2}}}=\dfrac{kg.m^3}{N.s^5}=\dfrac{kg.m^3}{(kg.m/s^2)(s^5)}=m^2/s^3$

For B: $\dfrac{\sqrt{(N.m.s)(N.m^2/kg^2)}}{(m/s)^{3/2}}=\dfrac{Ns^2}{kg}=\dfrac{(kg.m/s^2)(s^2)}{kg}=m$, it is the unit length.

For C: $\dfrac{\sqrt{(N.m.s)(N.m^2/kg^2)}}{(m/s)^{5/2}}=\dfrac{Ns^3}{m.kg}=\dfrac{(kg.m/s^2)(s^3)}{m.kg}=s$

For D: $\sqrt{\dfrac{(N.m.s)(m/s)}{N.m^2/kg^2}}=kg$

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Single Correct Hard Published on 18th 08, 2020
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