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?


ANSWER

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


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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