Question

If the velocity of light $c$, gravitational constant $G$, and Planck's constant $h$ be taken as fundamental units, the dimension of mass in the new system will be :

• A. $c^{1/2}{h}^{1/2}{G}^{1/2}$
• B. $c^{1/2}{h}^{1/2}{G}^{-1/2}$
• C. $c^{-3/2}{h}^{1/2}{G}^{1/2}$
• D. $c^{-3/8}{h}^{1/2}{G}^{1/2}$

Answer 1

Answer: (B)

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

Dimensions of velocity of Light c-- $L{T}^{-1}$

Dimensions of gravitational constant G -- $\frac{L^3}{M^1{T}^2}$

Dimensions of planck's constant h -- $\frac{M{L}^2}{T^1}$

Let a,b and c be powers of c,G and h respectively such that their product gives dimensions of mass.

$c^a{G}^b{h}^c=(L{T}^{-1}{)}^a\times {\left(\frac{L^3}{M^1{T}^2}\right)}^b\times {\left(\frac{M{L}^2}{T^1}\right)}^c=M^1{L}^0{T}^0$

on solving,

a=$1/2$, b=$-1/2$, c=$1/2$.

Hence, Option B is correct.