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Dimensional Analysis of Planck Length

  1. Aug 12, 2015 #1

    So in looking at black hole thermodynamics, I came across the equation

    ## l_p = \sqrt{G\hbar} ##

    But in doing a dimensional analysis of
    ## \sqrt{G\hbar} ##
    I get

    ## [\sqrt{G \hbar}] = \sqrt{ \frac{Nm^2}{kg^2} \frac{m^3}{kgs} } ##

    This obviously doesn't amount to a length. What am I doing wrong here?
  2. jcsd
  3. Aug 12, 2015 #2
    The planck length is equal to
    [tex] \sqrt { \frac {G \hbar} { c^3} } [/tex]

    You got the dimension of plancks constant wrong. A joule second is not m^3/(kg.s)

    You also need to convert newton to kg, meter and second.
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