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Help me solve this 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

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 ?

• Option 1)

$Gh^{^{2}}c^{3}$

• Option 2)

$G^{2}hc$

• Option 3)

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

• Option 4)

$\left ( \frac{Gh}{c^{3}} \right )^{1/2}$

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As we have learnt @1035

let planck length is

$L = K.G^{^{x}}h^{y}c^{z}$

$\left [M^{0}L^{1}T^{0} \right ] = \left [M^{-1}L^{3}T^{-2} \right ]^{x}\left [ML^{2}T^{-1} \right ]^{y}\left [LT^{-1} \right ]^{z}$

$= \left [ M^{-x+y}.L^{3x+2y+z}.T^{-2x-y-z} \right ]$

$\therefore -x+y=0 \: or\: x=y$

$\therefore -2x-y-z=0\: or\: 2x+y+z=0\: or \: z=-3x$

$\therefore 3x+2y+z=1 or 2x=1 or x=1/2$

$\therefore y=1/2$

$\therefore z=-3/2$

Planck Length = $\left [\frac{Gh}{c^{3}} \right ]^{1/2}$

Option 1)

$Gh^{^{2}}c^{3}$

Option 2)

$G^{2}hc$

Option 3)

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

Option 4)

$\left ( \frac{Gh}{c^{3}} \right )^{1/2}$

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