Question

The speed of light $(c)$, gravitational constant $(G)$ and Planck's constant $(h)$ are taken as fundamental units in a system. The dimensions of time in this new system should be

• A. $G^{1 / 2} h^{1 / 2} c^{-5 / 2}$
• B. $G^{-1 / 2} h^{1 / 2} c^{1 / 2}$
• C. $G^{1 / 2} h^{1 / 2} c^{-3 / 2}$
• D. $G^{1 / 2} h^{1 / 2} c^{1 / 2}$

Answer 1

Answer: (A)

According to the method of dimensional analysis the dimension of each term on both sides of an equation must be same.
Time $\propto c^{x} G^{y} h^{z}$
$\Rightarrow T=k c^{x} G^{y} h^{z}$
Putting the dimensions in above relation
$\Rightarrow \left[ M ^{0} L ^{0} T ^{1}\right]=\left[ LT ^{-1}\right]^{x}\left[ M ^{-1} L ^{3} T ^{-2}\right]^{y}\left[ ML ^{2} T ^{-1}\right]^{z}$
$\Rightarrow \left[ M ^{0} L ^{0} T ^{1}\right]=\left[ M ^{-y+z} L ^{x+3 y+2 z} T ^{-x-2 y-z}\right]$
Comparing the powers of $M , L$ and $T$
$-y+z=0 $ ... (i)
$x+3 y+2 z=0 $ ... (ii)
$-x-2 y-z=1$ ... (iii)
On solving Eqs. (i), (ii) and (iii)
On solving Eqs. (i), (ii) and (iii)
$x=-\frac{5}{2}, y=z=\frac{1}{2}$
Hence, dimensions of time are $\left[G^{1 / 2} h^{1 / 2} c^{-5 / 2}\right]$