Coulomb's Law

Electric charge is a fundamental property of certain particles, and we must describe it and the interactions caused by it between particles. Charge can be positive or negative with various magnitudes, and a charge exerts a force on another charge. The force between charges has the following properties:

1. Charges of the same polarity repel, those of different polarity attract.
2. The magnitude of the force is proportional to the product of the charge magnitudes, $q_1$ and $q_2$.
3. The magnitude of the force is inversely proportional to the square of the distance between the two charges, $r^2$.
4. The magnitude of the force is dependent on the medium in which the two charges are found. If we decide to represent this medium using a property of ‘permittivity’, $ϵ$, then the magnitude in inversely proportional to this. In a vacuum the medium is the electromagnetic field and the permittivity if the vacuum permittivity, ${ϵ}_0$.

Thus,

$$F_e\propto \frac{q_1{q}_2}{ϵr^2}$$

Experimentally it is known that this equation requires a constant of proportionality, $k$, to become equal. So,

$$F_e=k\frac{q_1{q}_2}{ϵr^2}$$

This equation represents the magnitude only; if we want to fully describe the electric force between charges, we must also account for direction by making this a vector quantity. We do this by adding a unit vector which sits at one charge and points directly away from the other charge. So finally,

$${\hat{F}}_e=k\frac{q_1{q}_2}{ϵr^2}{\hat{a}}_r$$

Note that if the charges are the same polarity, both positive or both negative, then the force on each charge will point away from the other. Likewise, if the charges are opposite in polarity, one positive and one negative, then the force vector will also turn out negative and the force will point towards the other charge. Thus we have represented the attraction and repulsion of like and unlike charges.