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
$\Rightarrow \left[M^0{L}^0{T}^l\right]={\left[L{T}^{-1}\right]}^x{\left[M^{-1}{L}^3{T}^{-2}\right]}^y{\left[M{L}^2{T}^{-1}\right]}^z$
$\Rightarrow \left[M^0{L}^0{T}^l\right]=\left[M^{y+z}{L}^{x+3y+2z}{T}^{-x-2y-z}\right]$
Comparing the powers of $M,L$ and $T$
$-y+z=0$(i)
$x+3y+2z=0$(ii)
$-x-2y-z=1$(iii)
On solving Eqs. (i), (ii)and (iii)
$x=-\frac{5}{2},y=z=\frac{1}{2}$
Hence, dimensions of time are $\left[G^{1/2}{h}^{1/2}{c}^{-5/2}\right]$