Physics - Unit & Dimension Question with Solution | TestHub

Let time,  $T\propto c^X{G}^Y{h}^Z$  $\Rightarrow T=k{c}^X{G}^Y{h}^Z$ Taking dimensions on both sides $[M^0{L}^0{T}^1]={[L{T}^{-1}]}^X{[M^{-1}{L}^3{T}^{-2}]}^Y{[M{L}^2{T}^{-1}]}^Z$ i.e., $[M^0{L}^0{T}^1]=[M^{-Y+Z}{L}^{X+3Y+2Z}{T}^{-X-2Y-Z}]$ Equating powers of $M,L$, and $T$, on both sides, we get – $-Y+Z=0\mathrm{...}(1)$  $X+3Y+2Z=0\mathrm{...}(2)$  $-X-2Y-Z=1\mathrm{...}(3)$ From equation (i)Adding (2) and (3)  $Y+Z=1$ or           $2Y=1\left(\because Z=Y\right)$  $\therefore Z=Y=\frac{1}{2}$ Putting these values in (2), we get – $X+\frac{3}{2}+1=0\mathrm{or}X=\frac{-5}{2}$ Hence,  $[T]=[G^{\frac{1}{2}}{h}^{\frac{1}{2}}{c}^{\frac{-5}{2}}]$