Question

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

• A. $h^{1/2}{C}^{1/2}{G}^{-1/2}$
• B. $h^{-1}{C}^{-1}G$
• C. $hC{G}^{-1}$
• D. $hCG$

Answer 1

Answer: (A)

$h^{1/2}{C}^{1/2}{G}^{-1/2}$

Speed of light $C=\left[{LT}^{-1}\right]$

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

planck's constant $h={ML}^2{T}^{-1}$

Let $M=C^x{G}^y{h}^z$

$\left[M\right]={\left[{LT}^{-1}\right]}^x{\left[M^{-1}{L}^3{T}^{-2}\right]}^y{\left[{ML}^2{T}^{-1}\right]}^z$

$\left[M\right]=\left[M^{-y+z}{L}^{x+3y+2z}{T}^{-x-2y-z}\right]$

$-y+z=1,x+3y+2z=0,-x-2y-z=0$

On solving, we get

$x=\frac{1}{2};y=\frac{-1}{2};z=\frac{1}{2}$

So, dimension of $M=\left[C^{1/2}{G}^{-1/2}{h}^{1/2}\right]$