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:

$\d \tau$ is an infinitesimal volume element

$\mathbf r$ is the position vector of $\d \tau$

$V$ 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.

Cartesian Form

The total electric charge on $\mathrm H$ carried by the electron can be expressed in Cartesian coordinates as:

\(\ds \int_{\text {all space} } \map {\rho_{\mathrm {el} } } {\mathbf r} \rd \tau\)

\(=\)

\(\ds \)

\(\ds \int_{-\infty}^\infty \int_{-\infty}^\infty \int_{-\infty}^\infty \map {\rho_{\mathrm {el} } } {\mathbf r} \rd x \rd y \rd z\)

\(=\)

\(\ds -\E\)

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.

$\blacksquare$

Sources

• 1990: I.S. Grant and W.R. Phillips: Electromagnetism (2nd ed.) ... (previous) ... (next): Chapter $1$: Force and energy in electrostatics: $1.3$ Electric Fields in Matter: $1.3.1$ The atomic charge density