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)

The correct options are
A M $\propto \sqrt{c}$
C L $\propto \sqrt{h}$
D L $\propto \sqrt{G}$

For $M:\left[M\right]=\left[h{]}^p\right[c{]}^q[G{]}^r$

$\left[M\right]=\left[M^1{L}^2{T}^{-1}{]}^p\right[L^1{T}^{-1}{]}^q[M^{-1}{L}^3{T}^{-2}{]}^r$

$\left[M\right]=\left[M{]}^{p-r}\right[L{]}^{2p+q+3r}[T{]}^{-p-q-2r}$

$\therefore p-r=1\dots \left(i\right)$

$2p+q+3r=0\dots \left(ii\right)$

$-p-q-2r=0\dots \left(iii\right)$

on solving (i), (ii), and (iii), we get

$p=\frac{1}{2},q=\frac{1}{2},p=-\frac{1}{2},$

$\therefore M\propto \sqrt{h}\sqrt{c}\frac{1}{\sqrt{G}}.$



For $L:\left[L\right]=\left[M{]}^{p-r}\right[L{]}^{2p+q+3r}[T{]}^{-p-q-2r}$

$p-r=0\dots \left(iv\right)$

$2pq+3r=1\dots \left(v\right)$

$-p-q-2r=0\dots \left(vi\right)$

On solving (iv), (v), and (vi), we get

$p=\frac{1}{2},q=-\frac{3}{2},r=\frac{1}{2}$

$\therefore L\propto \sqrt{h}\frac{1}{c^{\frac{3}{2}}}\sqrt{G}$