##### Coulomb's Law
ems.es.efo.cl

- FLS II 4-2
- Gr 2.1.2,4
- PM 1.4

- FLS II (4.9)
- Gr (2.1)
- PM (1.1)
- PS (3.1)
- W (2-3)

The force \(F_{t\leftarrow s}\) exerted by a point source charge \(q_s\) sitting at \({\bf r}_s\) on a point test charge \(q_t\) sitting at \({\bf r}_t\) is given by Coulomb's law,

\[ {\bf F}_{t\leftarrow s} = \frac{q_t q_s}{4\pi \varepsilon_0} \frac{{\bf r}_t - {\bf r}_s}{|{\bf r}_t - {\bf r}_s|^3} = \frac{q_t q_s}{4\pi \varepsilon_0} \frac{\hat{\bf r}_{ts}}{r_{ts}^2} \tag{Cl}\label{Cl} \]

In case you're wondering: the exact number in the denominator of the first equality is just \(4 \pi c^2\), so this equation really specifies what a Farad is.

One Farad has units \(F (= C/V) = s^4 A^2/kg~m^2\) in the SI system (the Ampère is a base SI unit; the Coulomb and the Volt are derived ones). Equivalently, \(F/m = C^2/N m^2\), but again the Newton \(N\) is a derived (not base) SI unit.

in which \(\varepsilon_0\), called the permittivity of free space (or alternately vacuum permittivity or sometimes electric constant), takes the value

\[ \varepsilon_0 \equiv \frac{1}{\mu_0 c^2} = \frac{1}{35 950 207 149.472 7056 \pi} F/m \\ \simeq 8.854 187 817 ... \times 10^{-12} F/m \]

in which \(\mu_0\) is the vacuum permeability (or alternately permeability of free space, permeability of vacuum or magnetic constant),

\[ \mu_0 = 1.25663706212(19)x10^{-6} H/m \]

with the henry \(H = kg~m^2 / s^2 A^2\) being the unit for inductance.

Info: the International System of Units

Throughout these notes we will use the International System of Units (SI). Units are perhaps less set in stone than you might think, because of their interdependence. For example, the speed of light is a measurable constant of nature, and therefore one must be careful when defining the metre and the second.

Since 1960, the speed of light is defined to be exactly \(299 792 458 ~m/s\). Since 1983, the metre is defined as the distance travelled by light in \(1/299 792 458 s\).

In the good old days (namely up to 2019), the vacuum permeability was defined to have the handy value \(\mu_0 = 4\pi \times 10^{-7} H/m\). This is no longer true.

The 2019 SI redefinition uses seven base units: second (s), metre (m), kilogram (kg), Ampere (A), kelvin (K), mole (mol) and candela (cd), all defined in terms of invariant constants of nature.

The vacuum permeability is thus no longer a defined constant and must be measured experimentally (it depends on the fine structure constant as \(\mu_0 = 2\alpha \frac{h}{e^2 c}\)). Its currently measured value is given in the main text.

Except where otherwise noted, all content is licensed under a Creative Commons Attribution 4.0 International License.

Author: Jean-Sébastien Caux

Created: 2022-03-24 Thu 08:42