Dimensional Analysis of Planck Length

[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?

 

[willem2]

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?

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.