Question

If the dimension of time are expressed as $G^{\alpha }{c}^{\beta }{h}^{\gamma }$, then the values of $\alpha$ and $\gamma$ are
[ $G$: Universal Gravitational Constant
$c$ : Speed of light
$h$: Planck's constant]

• A. $\frac{1}{2},\frac{1}{2}$
• B. $\frac{1}{2},-\frac{5}{2}$
• C. $-\frac{5}{2},\frac{1}{2}$
• D. $\frac{1}{2},-\frac{1}{2}$

Answer 1

The correct option is A $\frac{1}{2},\frac{1}{2}$

We know

$\left[G\right]=\left[M^{-1}{L}^3{T}^{-2}\right]$

$\left[c\right]=\left[L{T}^{-1}\right]$

$\left[h\right]=\left[M{L}^2{T}^{-1}\right]$

Now,

$T^1=\left[G{]}^{\alpha }\right[c{]}^{\beta }[h{]}^{\gamma }$

or,

$T^1=\left[M^{-1}{L}^3{T}^{-2}{]}^{\alpha }\right[L{T}^{-1}{]}^{\beta }[M{L}^2{T}^{-1}{]}^{\gamma }$

or,

$T^1=M^{-\alpha +\gamma }{L}^{3\alpha +\beta +2\gamma }{T}^{-2\alpha -\beta -\gamma }$
On comparing the power of $M,L$ and $T$ we have

$-\alpha +\gamma =0$

or,

$\gamma =\alpha$ ..................(i)

Again,

$3\alpha +\beta +2\gamma =0$

$3\alpha +2\alpha +\beta =0$ $[\because \alpha =\gamma ]$

or,

$\beta =-5\alpha$ ................(ii)

Again,

$-2\alpha -\beta -\gamma =1$

or,

$-2\alpha +5\alpha -\alpha =1$

or,

$2\alpha =1$

or,

$\alpha =\frac{1}{2}$

$\therefore \alpha =\gamma =\frac{1}{2}$ and $\beta =-\frac{5}{2}$

$\therefore \left[t\right]=\left[G{]}^{1/2}\right[c{]}^{-5/2}[h{]}^{1/2}$

(Option A)