Total Charge carried By Electron in Hydrogen Atom/Cartesian Form

Theorem

Consider an atom of hydrogen $\mathrm H$.

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

Follows directly.

$\blacksquare$

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