Coulomb's Law

The magnitude of electric force (Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle F} ) between two point charges is directly proportional to the product of the charges ($q_{1}$ and $q_{2}$) and inversely proportional to the square of the distance between them ($r$)

$F=k\,{\frac {\left|q_{1}q_{2}\right|}{r^{2}}}$

This equation will often be written in the following form for simplification purposes in the future.

$F={\frac {1}{4\pi \epsilon _{0}}}\,{\frac {\left|q_{1}q_{2}\right|}{r^{2}}}$

Constants

• $k={\frac {1}{4\pi \epsilon _{0}}}=(10^{-7}\ \mathrm {N} \cdot \mathrm {s} ^{2}/\mathrm {C} ^{2})c^{2}\approx 8.988\times 10^{9}\ \mathrm {N} \cdot \mathrm {m} ^{2}/\mathrm {C} ^{2}$
• $c=2.99792458\times 10^{8}\ \mathrm {m} /\mathrm {s}$ (the speed of light for above equation)
• $\epsilon _{0}\approx 8.854\times 10^{-12}\ \mathrm {C} ^{2}/(\mathrm {N} \cdot \mathrm {m} ^{2})$
• $e=1.6021765314\times 10^{-19}\ \mathrm {C}$ (the charge of an electron)

Example

An α particle is the nucleus of a helium atom:

• mass ($m$) = 6.64 × 10⁻²⁷ kg
• charge (Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q} ) = Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle +2e} = 3.2 × 10⁻¹⁹ C

Compare the force of the electric repulsion between two α particles with the force of gravitational attraction between them (Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle F_e/F_g} )

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align} F_e &= \frac{1}{4\pi\epsilon_0}\,\frac{q^2}{r^2} \\ F_g &= G\,\frac{m^2}{r^2} \\ \frac{F_e}{F_g} &= \frac{1}{4\pi\epsilon_0G}\,\frac{q^2}{m^2} = 3.1 \times 10^{35} \end{align} }

As stated in the footnotes, the gravitational force is negligible, hence the huge number from the resulting ratio.