Total Charge carried By Electron in Hydrogen Atom

Theorem
Consider an atom of hydrogen $\mathrm H$.

Then:
 * $\ds \int_V \map {\rho_{\mathrm {el} } } {\mathbf r} \rd \tau = -\E$

where:
 * $\rd \tau$ is an infinitesimal volume element
 * $\mathbf r$ is the position vector of $\rd \tau$
 * $V$ is a volume large enough to completely enclose $\mathrm H$
 * $\map {\rho_{\mathrm {el} } } {\mathbf r}$ is the electric charge density caused by the charge on the electron in the electron cloud at $\mathbf r$
 * $\E$ is the elementary charge.

Proof
The total electric charge on $\mathrm H$ carried by the electron is equal to the total charge on the electron.

By definition of the charge on the electron, this total is $-\E$.

The result follows.