Coulomb's Law

Similar to the gravitational force between two objects you'll find Coulomb's law easier to define and express the force between two charges.

Here we denote a net point charge by symbol $q$. 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 centre 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. Now consider two fixed 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).

Figure 1 Two point charges $q_1$ and $q_2$ separated by a distance $r$ exert force on each other.

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, $k=9.0\times {{10}^{9}}\frac{\text{N}{{\text{m}}^{\text{2}}}}{{{\text{C}}^{\text{2}}}}$).

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