Question

The characteristic distance at which quantum gravitational effects are significant, the Planck length, can be determined from a suitable combination of the fundamental physical constants $G,h$ and $c$. Which of the following correctly gives the Planck length ?

• A. $G{\hslash }^2{c}^3$
• B. $G^2\hslash c$
• C. $G^{1/2}{\hslash }^{2}c$
• D. ${\left(\frac{G\hslash }{c^3}\right)}^{1/2}$

Answer 1

Answer: (D)

Given Planck length $l$ can be determined from suitable combination of $G,\hslash$ and $c$. Therefore,
$l=f(G,h,c)={\left[M^{-1}{L}^3{T}^{-2}\right]}^x{\left[M{L}^2{T}^{-1}\right]}^y{\left[L{T}^{-1}\right]}^2=M^{-2}{M}^y{L}^{3x}{L}^{2y}{L}^z{T}^{-2x}{T}^{-y}{T}^z$
$\Rightarrow M^{\circ }{L}^1{T}^0=M^{x+y}{L}^{3x+2y+z}{T}^{-2x-y-z}$
Equating the exponents, we have
$-x+y=0\Rightarrow x=y$
$-2x-y-z=0$
$\Rightarrow -2x-x-z=0$
$\Rightarrow z=-3x$
$3x+2y+z=1$
$\Rightarrow 3x+2x-3x=1$
$\Rightarrow x=\frac{1}{2}$
$\Rightarrow x=\frac{1}{2},y=\frac{1}{2},z=\frac{-3}{2}$
Therefore, $l=G^{1/2}{h}^{1/2}{c}^{-3/2}=\sqrt{\frac{G\hslash }{c^3}}$