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 $\propto c^x G^y h^x$
$\Rightarrow T=k c^x G^y h^z$
Putting the dimensions in above relation
$\Rightarrow {\left[M^0 L^0 T^{ l }\right]=\left[L T^{-1}\right]^x\left[M^{-1} L^3 T^{-2}\right]^y\left[M L^2 T^{-1}\right]^z} $
$\Rightarrow {\left[M^0 L^0 T^{ l }\right]=\left[M^{y+z} L^{x+3 y+2 z} T^{-x-2 y-z}\right]}$
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)
$x=-\frac{5}{2}, y=z=\frac{1}{2}$
Hence, dimensions of time are $\left[G^{1 / 2} h^{1 / 2} c^{-5 / 2}\right]$