Question

The energy ($E$), angular momentum ($L$) and universal gravitational constant ($G$) are chosen as fundamental quantities. The dimensions of universal gravitational constant in the dimensional formula of planks constant ($h$) is :

• A. $0$
• B. $-1$
• C. $\frac{5}{3}$
• D. $1$

Answer 1

The correct option is A $0$

$h=G^{\alpha }{E}^{\beta }{L}^{\gamma }$

$(M\frac{L^2}{T^2}.T)={\left[\frac{M^{-1}L3}{T^2}\right]}^{\alpha }{\left[\frac{M{L}^2}{T^2}\right]}^{\beta }.{\left[M\frac{L^2}{T}\right]}^{\gamma }$

$\left[M{L}^2{T}^{-1}\right]=M^{-\alpha +\beta +\gamma }{L}^{3\alpha +2\beta +2\gamma }{T}^{-2\alpha -2\beta -\gamma }$
$\Rightarrow -\alpha +\beta +\gamma =1$
$\Rightarrow 3\alpha +2\beta +2\gamma =2$
$\Rightarrow \alpha =0=\beta$
$-2\alpha -2\beta -\gamma =-1$
$\gamma =1$
Since $\alpha =0$,
G in dimensional formula for h is 0.