Q. If the velocity of light $c,$ gravitational constant $G$ and Planck’s constant $h$ are chosen as fundamental units, the dimensions of length $L$ in the new system is
A) $hcG^{- 1}$
B) $\lbrack h^{1/2}c^{1/2}G^{- 1/2}\rbrack\$
C) $\left\lbrack hc^{- 3}G^{1} \right\rbrack$
D) $\left\lbrack h^{1/2}c^{- 3/2}G^{1/2} \right\rbrack$
Explanation
Let $\ \ \ \ \ \ \ \ \ \ \ L = \left\lbrack h^{a}c^{b}G^{c} \right\rbrack$ $\therefore\ \ \ \ \ \ \ \ \ \ \left\lbrack L^{'} \right\rbrack = \left\lbrack ML^{2}T^{- 1} \right\rbrack^{a}\left\lbrack LT^{- 1} \right\rbrack^{b}\left\lbrack M^{- 1}L^{3}T^{- 2} \right\rbrack^{c}$ $\Rightarrow \ \ \ \ \ \ \ \ \ \ \ \ a = \frac{1}{2},\ b = - \frac{3}{2},\ c = \frac{1}{2}$ Hence, $L = \left\lbrack h^{1/2}c^{- 3/2}G^{1/2} \right\rbrack\$
Reviewed by Dr. Abubacker
Physics Educator & NEET/JEE Specialist
Ph.D. in Physics with 19+ years of teaching experience.