# Coulomb's Law Between Electric Charges

If static electricity is use to charge two bodies, they will either show repulsion or attraction towards each other depending on what the nature of their charge is. This statement was first observed by the Greek philosopher Thales of Miletus in 600 BC. This remained as an observation without any mathematical evidence of this force that caused the repulsion attraction between objects until a French Physicist Charles Augustin de Coulomb in 1785 coined the tangible relationship in mathematical form in between two bodies that have been electrically charged and published an equation for the force causing the bodies to attract or repel each other. The relation is now known as the Coulomb’s Law.

## Coulomb’s Law

Consider two point electric charges with charge values q1 and q2 separated by a distance r.

Coulomb’s law establishes that,

The force exerted by two charges between each other is proportional to the product of the charge values and is inversely proportional to the square of the distance between them.

$$\overrightarrow{F} ∝ \frac{q_1q_2}{r^2}$$

$$\overrightarrow{F}$$= $$K \frac{q_1q_2}{r^2}$$

The constant of proportionality K is determined to be $$K$$ = $${1}{4πε_0}$$

Thus,

$$\overrightarrow{F}$$ = $$\frac{1}{4\piε_0} \frac{q_1q_2}{r^2} \hat{r}$$

Also, $$\frac{1}{4πε_0}$$ = $$8.99 × {10}^{9} \frac{Nm^2}{C^2}$$

And, $$ε_0$$ = $$8.85 × {10}^{-12} \frac{C^2}{Nm^2}$$

$$ϵ_0$$ is known as the permittivity constant of free space. Note that this equation provides the force of only charges in vacuum.

The sign of force determines the whether the force is attractive or repulsive. Since, unlike charges attract, the sign is given as negative. A positive sign of force suggests a repulsive force since like charges repel. The direction of force is along the line separating the charges, i.e., it is along the vector $$\overrightarrow{r}$$.

Note that this inverse square law has an astounding similarity to the Universal Law of Gravitation given by Newton except that gravitational force is always attractive.

$$\overrightarrow{F}$$ = $$G \frac{m_1m_2}{r^2} \hat{r}$$

This inverse square law implies that the force diminishes rapidly with increasing distance. Another profound implication is that every charge in the universe exerts a force on every other charge no matter how far it is! Only the amount of force exerted is exceedingly small for far charges.

We can build upon this concept to calculate the force exerted due to multiple charges by adding the individual contributions by pair of charges vector allies.

#### Practise This Question

A charge Q is divided into two parts of q and Q-q. If the coulomb repulsion between them when they are separated is to be maximum, the ratio of Qq should be