# What does the Planck-like charge $Q=\sqrt{\frac{\pi}{c\hbar \varepsilon_0}}$ represent?

While experimenting something with equations I got a equation of charge which is not Planck's charge, dimensionally it is correct. I used dimensions method to get this. $$Q=\sqrt{\frac{\pi}{c\hbar \varepsilon_0}}$$ where $$c$$ is speed of light and $$\varepsilon_0$$ is permittivity of free space. This is lot more different that of Planck charge but contains the same constants as that of Planck.

For one who want to know how I found it: I deleted the Coulomb constant and dimensionally found out permittivity inputing the entire Coulomb constant. Its like taking square of Coulomb's constant. I know this is not Planck charge because while finding them the dimension changed.

What is this "charge"? Is it significant?

## 1 Answer

Up to a numerical factor of order 1, it is the fundamental electric charge $$e$$ (the magnitude of the electron’s charge). Read about the fine-structure constant to see why.

Correction... it would be if your dimensions were correct, but they aren’t. The expression on the right-hand is not a charge but an inverse charge.

Another correction... The Planck charge is $$\sqrt{4\pi\epsilon_0\hbar c}$$ or $$e/\sqrt{\alpha}$$, so if you inverted the right hand side you would be close to the Planck charge, which is roughly an order of magnitude more than the elementary charge.