Question

Planck's constant $h$, speed of light $c$ and gravitational constant $G$ are used to form a unit of length $L$ and a unit of mass $M$. Then the correct option(s) is/are

• A. $M\propto \sqrt{c}$
• B. $M\propto \sqrt{G}$
• C. $L\propto \sqrt{h}$
• D. $L\propto \sqrt{G}$

Answer 1

Answer: (A), (C), (D)

$M\propto \sqrt{c}$
$L\propto \sqrt{h}$
$L\propto \sqrt{G}$

$h=M{L}^2{T}^{-1}$

$c=L{T}^{-1}$

$G=M^{-1}{L}^3{T}^{-2}$

$L=h^a{c}^b{G}^d$

$M^0{L}^1{T}^0={M^{a}L}^{2a}{T}^{-a}{L}^b{T}^{-b}{M}^{-d}{L}^{3d}{T}^{-2d}$

Equating coefficients of M, L and T

$a-d=0,2a+b+3d=1,-a-b-2d=0$

$a=\frac{1}{2},d=\frac{1}{2},b=\frac{-3}{2}$

$L\alpha \sqrt{h},L\alpha \sqrt{G}$

Similarly using dimenitional analysis for second case

$M=h^a{c}^b{G}^d$

$M^1{L}^0{T}^0={M^{a}L}^{2a}{T}^{-a}{L}^b{T}^{-b}{M}^{-d}{L}^{3d}{T}^{-2d}$

$a-d=1,2a+b+3d=0,-a-b-2d=0$

$a=\frac{1}{2},d=\frac{-1}{2},b=\frac{1}{2}M\alpha \sqrt{h},M\alpha 1/\sqrt{G},M\alpha \sqrt{c}$

Correct options are A, C and D.