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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}

 
Answers (2)
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V Vakul

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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