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)

$G^{1/2}{h}^{1/2}{c}^{-5/2}$

Let time, $T\alpha c^x{G}^y{h}^z$

$\Rightarrow T=k{c}^x{G}^y{h}^z$

Taking dimensions on both sides,

$\left[M^0{L}^0{T}^1\right]={\left[L{T}^{-1}\right]}^x{\left[M^{-1}{L}^3{T}^{-2}\right]}^y{\left[M{L}^2{T}^{-1}\right]}^z$

i.e.,

$\left[M^0{L}^0{T}^1\right]=\left[M^{-y+z}{L}^{x+3y+2z}{T}^{-x-2y-z}\right]$

Equating power of M, L, T on both sides, we get

$-y+z=0........\left(1\right)$

$x+3y+2z=0.....\left(2\right)$

$-x-2y-z=1.......\left(3\right)$

From $\left(1\right)\Rightarrow z=y$

Adding $\left(2\right)$ and $\left(3\right)\Rightarrow y+z=1$

or $2y=1$ (From (1)) i.e., $y=\frac{1}{2}$

$\therefore z=y=\frac{1}{2}$

Putting these values in $\left(2\right)$ we get

$x+\frac{3}{2}+1=0$ or $x=\frac{-5}{2}$

Hence, $\left[T\right]=\left[G^{1/2}{h}^{1/2}{c}^{-5/2}\right]$