# Electric charges

#### Coulomb's law

Electric charge is a property of matter known since ancient times. Thales was probably the first one to point out its effects, making observations when rubbing amber rods against a cloth (“amber” in ancient greek was “ηλεχτρον” (Elektron), hence “electricity”).

Various experiments gave evidence of the existence of two types of electric charge, named positive and negative, that interacted with each another by exerting forces. Matter generally appears electrically neutral, and this is due to the fact that the internal charges are distributed in (approximately) an identical way between positive and negative, therefore they cancel out.

The first successful attempt to accumulate electric charges takes place in the 17th century, thanks to van Musschenbroek's invention of a sort of capacitor, called Leyden jar. This is a device consisting of a glass bottle, both internally and externally covered with metal foil, and the internal foil is connected via a conductor to an electric generator. The two foils are the electrodes and the glass is the the dielectric (insulator).

By the end of the 18th century, the French physicist Charles-Augustin de Coulomb experimentally derived a law for the force of electrostatic attraction exerted between two point charges ${\displaystyle q_{1}}$ and ${\displaystyle q_{2}}$ , separated by a distance ${\displaystyle r}$:

${\displaystyle {\underline {F_{e}}}=k{\frac {q_{1}q_{2}}{r^{2}}}{\underline {u_{r}}}\ .}$

Let's briefly comment upon the law above stated: the electric force exerted by ${\displaystyle q_{1}}$ on ${\displaystyle q_{2}}$ is directly proportional to the product of the two charges (sign included), and inversely proportional to the square of the distance between the charges. ${\displaystyle {\underline {u_{r}}}}$ is the unit vector which has the same direction as the line from ${\displaystyle q_{1}}$ to ${\displaystyle q_{2}}$, therefore the direction of the force is determined by the sign of the product ${\displaystyle q_{1}q_{2}}$. This is consistent with the fact that opposite charges attract, whereas like charges repel. The unit of measurement used for the electric charge is the Coulomb (C ).

The constant of proportionality is ${\displaystyle k\approx 9\cdot 10^{9}{\frac {Nm^{2}}{C^{2}}}}$, however, for convenience, it is best to introduce a new constant (the reason why it is more convenient will be clear later on):

${\displaystyle \varepsilon _{0}={\frac {1}{4\pi k}}\approx 8,85\cdot 10^{-12}{\frac {C^{2}}{Nm^{2}}}\ .}$
Thus, Coulomb's law is rearranged as:
${\displaystyle {\underline {F_{e}}}={\frac {1}{4\pi \varepsilon _{0}}}{\frac {q_{1}q_{2}}{r^{2}}}{\underline {u_{r}}}\ .}$

The principle of superposition is valid. In particular, charge density is denominated as ${\displaystyle \rho }$ for a charge distributed on a volume, σ for a surface and λ for a line.

The generalised formula for Coulomb's law is therefore the following:

{\displaystyle {\begin{aligned}{\underline {F_{e}}}&={\frac {q}{4\pi \varepsilon _{0}}}\sum _{i=1}^{N}q_{i}{\frac {\left({\underline {r}}-{\underline {r_{i}}}\right)}{\left|{\underline {r}}-{\underline {r_{i}}}\right\vert ^{3}}}++{\frac {q}{4\pi \varepsilon _{0}}}\left\lbrack \int _{V}^{}{\frac {\rho \left({\underline {r^{'}}}\right)\left({\underline {r}}-{\underline {r^{'}}}\right)}{\left|{\underline {r}}-{\underline {r^{'}}}\right\vert ^{3}}}\,d{V^{'}}\right.\\&\left.+\int _{S}^{}{\frac {\sigma \left({\underline {r^{'}}}\right)\left({\underline {r}}-{\underline {r^{'}}}\right)}{\left|{\underline {r}}-{\underline {r^{'}}}\right\vert ^{3}}}\,d{S^{'}}+\int _{l}^{}{\frac {\lambda \left({\underline {r^{'}}}\right)\left({\underline {r}}-{\underline {r^{'}}}\right)}{\left|{\underline {r}}-{\underline {r^{'}}}\right\vert ^{3}}}\,d{l^{'}}\right\rbrack \ .\end{aligned}}}

Among the most interesting particles in the electrical phenomena, there are the protons (positively charged) and the electrons (negatively charged). A very successful experiment to measure their charge was carried out by Millikan: he managed to find a really good approximation for the value of the charge of an electron.

All experiments so far have showed that electric charge is always conserved: even when some charge is produced, it is always accompanied by the production of the same amount of charge of the opposite sign, so that overall, the total charge remains unchanged.