Question

If the velocity of light $C$, gravitational constant $G$ and Planck's constant $h$ are chosen as fundamental units, then the dimension of mass in new system is

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

Answer 1

The correct option is C $\left[c^{1/2}{G}^{-1/2}{h}^{1/2}\right]$

According to the question mass,
$m\propto c^{\alpha }{G}^{\beta }{h}^{\gamma }$
by substituting the dimensions,
$\left[M\right]={\left[L{T}^{-1}\right]}^a{\left[M^{-1}{L}^3{T}^{-2}\right]}^{\beta }{\left[M{L}^2{T}^{-1}\right]}^{\gamma }$
Comparing the power on both sides,
$-\beta +\gamma =1$
$\alpha +3\beta +2\gamma =0$
$-\alpha -2\beta -\gamma =0$
by solving above equation, we get
$\alpha =\frac{1}{2}$, $\beta =-\frac{1}{2}$ and $\gamma =\frac{1}{2}$
so,
$m\propto c^{\frac{1}{2}}{G}^{-\frac{1}{2}}{h}^{\frac{1}{2}}$
so, dimensional formula of mass
$=\left[c^{\frac{1}{2}}{G}^{-\frac{1}{2}}{h}^{\frac{1}{2}}\right]$

Final answer: (c)