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. $\left[G^{\frac{1}{2}}{h}^{\frac{1}{2}}{c}^{-\frac{5}{2}}\right]$
• B. $\left[G^{\frac{-5}{2}}{h}^{\frac{1}{2}}{c}^{\frac{1}{2}}\right]$
• C. $\left[G^{\frac{1}{2}}{h}^{\frac{1}{2}}{c}^{\frac{-3}{2}}\right]$
• D. $\left[G^{\frac{1}{2}}{h}^{\frac{1}{2}}{c}^{\frac{5}{2}}\right]$

Answer 1

The correct option is A $\left[G^{\frac{1}{2}}{h}^{\frac{1}{2}}{c}^{-\frac{5}{2}}\right]$

Here, [c] = $L{T}^{-1}$
$\left[G\right]=M^{-1}{L}^3{T}^{-2}$
$\left[h\right]=M^1{L}^2{T}^{-1}$
Let, $t\propto c^x{G}^y{h}^z$
By substituting the dimensions of each quantity in both the sides,
$T=\left(L{T}^{-1}{)}^x\right(M^{-1}{L}^3{T}^{-2}{)}^y(M{L}^2{T}^{-1}{)}^z$
By equating the power of M, L, T in both the sides,
$-y+z=0$
$x+3y+2z=0$
$-x-2y-z=1$
By solving the above equation,
We get, $x=\frac{-5}{2},y=\frac{1}{2},z=\frac{1}{2}$
So, $\left[t\right]=\left[c^{\frac{-5}{2}}{G}^{\frac{1}{2}}{h}^{\frac{1}{2}}\right]$