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

Answer 1

Answer: (A), (B), (D)

$M\propto \sqrt{c}$

$G=M^{–1}L3T^{–2}$

$c=L{T}^{–1}$

$h=M{L}^2{T}^{–1}$

$L=\left[Gh{]}^a\right[c{]}^b$

$L=\left[L^5{T}^{–3}{]}^a\right[LT–1{]}^b$

$\Rightarrow –3a–b=0$

$5a+b=1$

$2a=1$

$a=\frac{1}{2}$

$\Rightarrow \left[L\right]=\left[G{]}^{\frac{1}{2}}\right[h{]}^{\frac{1}{2}}[c{]}^{\frac{-3}{2}}$

i.e, $L\times \sqrt{G},L\propto \sqrt{h},L\propto \frac{1}{c\sqrt{c}}$

Now, $M=\left[G{]}^a\right[c{]}^b[h{]}^c$

$\Rightarrow M=\left[M^{–1}{L}^3{T}^{–2}\right]a\left[L{T}^{–1}{]}^b\right[M{L}^2{T}^{–1}{]}^c$

$\Rightarrow 1=–a+c$

$\Rightarrow a=c–1$

Also, $3a+b+2c=0$

$\Rightarrow b+5c=3$

Also, $2a+b+c=0\Rightarrow 3c+b=2$

$\Rightarrow c=\frac{1}{2}a=\frac{-1}{2}b=\frac{1}{2}$

$\Rightarrow M\propto \frac{1}{\sqrt{G}},M\propto \sqrt{c},M\propto \sqrt{h}$