# Dimensional Analysis of Planck Length

1. Aug 12, 2015

### noahcharris

Hello!

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. Aug 12, 2015

### willem2

The planck length is equal to
$$\sqrt { \frac {G \hbar} { c^3} }$$

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.