Question

If the velocity of light $(c)$, gravitational constant $(G)$ and Planck’s constant $(h)$ are chosen as fundamental units, then the dimensions of mass in the new system is

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

Answer 1

Answer: (C)

Let $m\propto c^x{G}^y{h}^z$ or $m=K{c}^x{G}^y{h}^z$
By substituting the dimensions of each quantity in both sides, we get
$\left[M^1{L}^0{T}^0\right]=K{\left[L{T}^{-1}\right]}^x{\left[M^{-1}{L}^3{T}^{-2}\right]}^y{\left[M{L}^2{T}^{-1}\right]}^z$
$=\left[M^{-y+z}{L}^{x+3y+2z}{T}^{-x-2y-z}\right]$
By equating the powers of $M$, $L$ and $T$ on both sides, we get
$-y+z=1$, $x+3y+2z=0$, $-x-2y-z=0$
By solving above three equations, $x=1/2$, $y=-1/2$ and $z=1/2$.
$\therefore m\propto c^{1/2}{G}^{-1/2}{h}^{1/2}$