Planck Length

#Physics
The length at which measurements of positions of particles smaller than this will create a blackhole

$\displaystyle l_{P}=\sqrt{ \frac{\hbar G}{c^{3}} }\cong \mathrm{1.616\cdot 10^{-35}m}$

Black Hole Derivation

$$
\begin{align}
\Delta x \Delta p & \sim\hbar\\\Delta p & \sim \frac{\hbar}{L} \\E & =pc \\&=\frac{\hbar c}{L} \\R_{s} & =\frac{2Gm}{c^{2}},E=mc^{2}\rightarrow m=\frac{E}{c^{2}}=\frac{\hbar}{Lc} \\&= \frac{2G\hbar}{c^{3}L},\text{ let }R_{s}=L \\L^{2} & =\frac{2G\hbar}{c^{3}} \\l_{P} & \sim\sqrt{ \frac{\hbar G}{c^{3}}}
\end{align}
$$

• Start with Heisenberg's Uncertainty Principle