# Coulomb's Law

The Coulomb's law allows us to understand the interaction between two or more net charges. The term *net charge* tells you that the isolated charge has it's own electric field and it is not neutralized.

We consider *gravitational pull* which tells us that all objects on the Earth's surface are being pulled by the force of gravity. It means that there is a force pulling all objects towards the center of the Earth and we call such a force per unit mass of a body as the *gravitational field*.

The gravitational field of a body is the gravitational force it exerts per unit mass of other bodies inside the influence of its field. Just like the Earth has a gravitational field which has an influence to all other objects inside its field, a fixed charge also has an *electric field* around it which also has an influence to other charges if present inside its electric field.

And similar to the gravitational field electric field of a point charge is the electric force it exerts per unit charge. Every charge produces an electric field around it which exerts a force on other charges inside its electric field.

We consider two fixed isolated non-neutralized charges ${{q}_{1}}$ and ${{q}_{2}}$ separated by a distance $r$ not very far from each other. The electric field of charge $q_1$ exerts force on charge $q_2$ and the electric field of charge $q_2$ exerts force on charge ${{q}_{1}}$.

According to Newton's third law the force of charge ${{q}_{1}}$ on ${{q}_{2}}$ ($\vec F_{\text{1 on 2}}$) is exactly equal and opposite to the force of $q_2$ on $q_1$ ($\vec F_{\text{2 on 1}}$). This is the mutual interaction of charges (see Figure 1).

Now what do we know about the force between the two fixed charges? We have Coulomb's law to get the idea of the force between two fixed charges which tells us that the force between any two fixed charges $q_1$ and $q_2$ is directly proportional to the product of $q_1$ and $q_2$ and inversely proportional to the square of distance between them. That means $F \propto {q_1}{q_2}$ and $F \propto \frac{1}{{{r^2}}}$. Therefore, according to Coulomb's law

\[F = k\frac{{{q_1}{q_2}}}{{{r^2}}} \tag{1} \label{1}\]

where $k$ is the proportionality constant and $k=1/4\pi\epsilon_0$. Here $\epsilon_0$ is another constant and its value is $\epsilon_0 = 8.85\times{10^{ - 12}}\frac{{{{\rm{C}}^{\rm{2}}}}}{{{\rm{N}}{{\rm{m}}^{\rm{2}}}}}$ in three significant figures. The value of $k$ is approximately $k = 9.0 \times {10^9}\frac{{{\rm{N}}{{\rm{m}}^{\rm{2}}}}}{{{{\rm{C}}^{\rm{2}}}}}$ but its value is $k = 8.987551787 \times {10^9}\frac{{{\rm{N}}{{\rm{m}}^{\rm{2}}}}}{{{{\rm{C}}^{\rm{2}}}}}$ in more significant figures. We use the approximate value in calculations (that is, $k=9.0\times {{10}^{9}}\frac{\text{N}{{\text{m}}^{\text{2}}}}{{{\text{C}}^{\text{2}}}}$).

The Coulomb's law determines the force between two net charges but this force is the result of mutual interaction of both charges not of one single charge. It's like the stretched spring where these charges are fixed at opposite ends, and the force spring applies on them is the force between these charges. In case of the familiar Earth-body system, in similar gravitational force derivation we derived, the force was due to both the mass of the Earth and the mass of the body. The Earth is incredibly bigger than the point mass, the effect of the point mass is not seen.