Everything in the universe attracts every other thing kept around it. The universal law of Gravitation gave us an idea regarding the kind of interaction between macroscopic and macroscopic bodies. But what happens when we reach the subatomic level? What forces act between electrons or protons? What are the forces responsible for ensuring the stability of an atom? These are electrostatic forces. Coulomb’s Law helps us determine the magnitude of these electric forces between two charged bodies.

The idea behind Coulomb’s law dates back to 600 BC when Greek philosopher Thales of Miletus first put it forward. He noticed that when two bodies charged with static electricity are brought close, they will either get attracted or repulsed by each other, depending upon the nature of their charges.

He just quoted the theory, but this theory was established into mathematical relation by Charles Augustin de Coulomb, a French physicist. He described the relation between two electrically charged bodies in terms of distance and charge, and this relation became popularly known as the “Coulomb’s Law”. Let us learn in detail about this law.

What is Coulomb’s Law?

Coulomb’s law defines the relationship between the electrostatic force of attraction and repulsion between two electrically charged bodies. According to this Law, the magnitude of the electrostatic force between two charged bodies is:

1. Directly proportional to the product of the magnitude of charges on the two bodies.
2. Inversely proportional to the square of the distance between the centres of the charged bodies.

Coulomb’s law acts along the line joining the centres of the two charged bodies. Since the relation describes an inverse square relation between force and distance between the two charged bodies, that is why the law is also referred to as Coulomb’s inverse-square law. The relation derived from Coulomb’s Law is quite similar to the gravitational force acting between two massive bodies; that is why Coulomb’s Law is considered as an electrical analogue of Newton’s Universal Law of Gravitation.

This law gave a perfect estimate of the force acting between two point charges. In physics, the word “Point charge” is used to define extremely small bodies. This means that the linear size of the charged bodies is very small, especially in comparison to the distance between the two bodies. Thus, by considering these charged bodies as point charges, it becomes easier for us to calculate the magnitude of electrostatic forces acting between them.

Understanding Coulomb’s Law

From Coulomb’s Law, we can conclude that charges of the same sign will push each other way due to the forces of repulsion acting between them. In contrast, charges with opposite polarities will pull each other closer due to the force of attraction acting between them. This can be stated as like charges repel each other and unlike charges attract each other.

To visualise this, imagine that there are two equal and opposite charges separated by a small distance. The two charges will attract each other. Now, if the two charges are brought closer, the force of attraction between the two will get stronger.
If the charges are moved apart, the force of attraction between the two will decrease. Thus, we can say that the force between two charged bodies varies inversely with the distance between the two charges. If \(d\) is the distance between the charged bodies, then according to Coulomb’s Law, the electrostatic force between them varies as:
\(F \propto \frac{1}{{{d^2}}}\)
Keeping these charges fixed, if the magnitude of the charge is now increased, the force of attraction between the charges increases and on decreasing the magnitude of these charges, we find that the force of attraction between the charges decreases. From Coulomb’s Law, if \(q_1\) and \(q_2\) be the magnitude of the charges, then:
\(F \propto {q_1}{q_2}\)
This force is affected by the medium in which the charges are kept. If \(\varepsilon \) represents the property of a medium, then according to Coulomb’s Law,
\(F \propto \frac{1}{\varepsilon }\)

Learn about Electrostatic Force here

Derivation of Coulomb’s Formula

Coulomb’s Law statement helps us understand the relationship between charge and distance and how it influences the electrostatic force (i.e. the electric force between charged bodies at rest). This force is also known as Coulomb’s force. To calculate the expression of Coulomb’s force, Let that there be two charges \(Q_1\) and \(Q_2\), if these two charges are kept at a distance \(r\) from each other, the force of attraction/repulsion between them is \(F\). Then:
\(F ∝ Q_1 Q_2\) ………..(i)
\(F \propto \frac{1}{{{r^2}}}\) …………(ii)
Combine equations (i) and (ii), we get
\(F \propto ({Q_1}{Q_2})/{r^2}\)
\(F = \frac{{k{Q_1}{Q_2}}}{{{r^2}}}\)
Here, \(k\) is the constant of proportionality.

\(k = \frac{1}{{4\pi {\varepsilon _0}}}\)
Thus,
\(F = \frac{1}{{4\pi {\varepsilon _0}}}.\frac{{{Q_1}{Q_2}}}{{{r^2}}}\)
Where, \(\varepsilon_0\) represents the absolute permittivity of the space or vacuum.
In terms of the S.I unit, the value of \({\varepsilon _0} = 8.85 \times {10^{ – 12}}\frac{{{{\text{C}}^2}}}{{{\text{N}}{{\text{m}}^2}}}\)
Thus,
\(k = \frac{1}{{4 \times 3.14 \times 8.85 \times {{10}^{ – 12}}\frac{{{{\text{C}}^2}}}{{{\text{N}}{{\text{m}}^2}}}}}\)
\(k = 9 \times {10^9}\,{\text{N}}{{\text{m}}^2}/{{\text{C}}^2}\)
Using the value of \(k\), the value of Coulomb’s force between two charges can be given as:
\(F = 9 \times {10^9} \times \frac{{{Q_1}{Q_2}}}{{{r^2}}}\;{\text{N}}\)
If the charges instead of vacuum are not kept in a medium like glass or water, the expression for Coulomb’s force acting between the two charges becomes:
\(F = \frac{{{Q_1}{Q_2}}}{{4\pi \varepsilon {r^2}}}\)
Where \(\varepsilon\) is the permittivity of the medium in which charges are present, its value is constant for a given medium.
The permittivity of a medium is related to absolute permittivity as \(\varepsilon = {\varepsilon _0}{\varepsilon _r}\)
Where, \({\varepsilon _r}\) is the relative permittivity of the medium.

Vector Form of Coulomb’s Law

We know that all physical quantities can be categorised into two Scalars or Vectors. Scalars are quantities with the only magnitude, while vectors are quantities with both direction and magnitude. Force is a vector quantity. Thus, it will be represented by an arrow over it.
To write the expression for Coulomb’s Law in its vector form, let there be two charges \(Q_1\) and \(Q_2\), such that \(\overrightarrow {{r_1}}\) and \(\overrightarrow {{r_2}}\) represent the position vectors of the two charges, respectively. The two charges will exert electrostatic forces on each other. Let \(\overrightarrow {F_{12}}\) be the force exerted by the charge \(Q_1\) on \(Q_2\) and \(\overrightarrow {F_{21}}\) be the force exerted by the charge \(Q_2\) on \(Q_1\). Suppose the corresponding vector from \(Q_1\) to \(Q_2\) is given by \(\overrightarrow {r_{21}}.\)

Thus, using triangle law,
\(\overrightarrow {{r_{21}}} = \overrightarrow {{r_2}} – \overrightarrow {{r_1}} \)
The direction of position vector from \(\overrightarrow {{r_1}} \) to \(\overrightarrow {{r_2}} \) and \(\overrightarrow {{r_2}} \) to \(\overrightarrow {{r_1}} \), can be given as:
\(\widehat {{r_{21}}} = \frac{{\overrightarrow {{r_{21}}} }}{{\left| {\overrightarrow {{r_{21}}} } \right|}}\)
\(\widehat {{r_{12}}} = \frac{{\overrightarrow {{r_{12}}} }}{{\left| {\overrightarrow {{r_{12}}} } \right|}}\)
Therefore, the force acting on the charge \(Q_1\) due to \(Q_2\), in vector form can be given as:
\(\overrightarrow {{F_{21}}} = \frac{1}{{4\pi {\varepsilon _0}}}.\frac{{{Q_1}{Q_2}}}{{{{\left| {\overrightarrow {{r_{21}}} } \right|}^2}}}\widehat {{r_{21}}}\)
Or,
\(\overrightarrow {{F_{21}}} = \frac{1}{{4\pi {\varepsilon _0}}}.\frac{{{Q_1}{Q_2}}}{{{{\left| {\overrightarrow {{r_{21}}} } \right|}^3}}}\overrightarrow {{r_{21}}} \)
Or in general,
\(\overrightarrow F = \frac{1}{{4\pi {\varepsilon _0}}}.\frac{{{Q_1}{Q_2}}}{{{{\left| {\overrightarrow r } \right|}^3}}}\overrightarrow r \)
The above equation is the vector form of Coulomb’s Law.
Here the polarities of both charges are the same. Thus, the two charges will repel each other. This means that \(\overrightarrow {{F_{12}}} \) is the repulsive force exerted by the charge \(Q_1\) on \(Q_2\) and \(\overrightarrow {{F_{21}}} \) is the repulsive force exerted by the charge \(Q_2\) on \(Q_1.\)
From above, the position vector from \(Q_1\) to \(Q_2\) is,
\(\overrightarrow {{r_{21}}} = \overrightarrow {{r_2}} – \overrightarrow {{r_1}} \)
The position vector from \(Q_2\) to \(Q_1\) is,
\(\overrightarrow {{r_{12}}} = \overrightarrow {{r_1}} – \overrightarrow {{r_2}} \)
Thus,
\(\overrightarrow {{r_{21}}} = – \overrightarrow {{r_{12}}} \)
Thus,
\(\overrightarrow {{F_{21}}} = \frac{1}{{4\pi {\varepsilon _0}}}.\frac{{{Q_1}{Q_2}}}{{{{\left| {\overrightarrow {{r_{21}}} } \right|}^3}}}\overrightarrow {{r_{21}}} = – \frac{1}{{4\pi {\varepsilon _0}}}.\frac{{{Q_1}{Q_2}}}{{{{\left| {\overrightarrow {{r_{12}}} } \right|}^3}}}\overrightarrow {{r_{12}}} \) ………..(1)
\(\overrightarrow {{F_{12}}} = \frac{1}{{4\pi {\varepsilon _0}}}.\frac{{{Q_1}{Q_2}}}{{{{\left| {\overrightarrow {{r_{12}}} } \right|}^3}}}\overrightarrow {{r_{12}}} \) ………..(2)
From equations (1) and (2).
\(\overrightarrow {{F_{21}}} = – \overrightarrow {{F_{12}}} \)
The force on the first charge due to the second charge and the force on the second charge due to the first charge are in opposite directions and equal in magnitude. Thus, Coulomb’s Law upholds Newton’s third law of motion, stating that every action has an equal and opposite reaction.

Superposition Principle:

For an assembly of charges \({Q_1},\,{Q_2},\,{Q_3},…\) the force on any charge, say \({Q_1},\,\) is the vector sum of the force on \({Q_1}\) due to \({Q_2}\) due to \({Q_3},\) and so on. For each pair, the force is given by the Coulomb’s law for two charges.

The Relative Permittivity of a Medium

The relative permittivity of the medium \({\varepsilon _r}\) can be given as
\({\varepsilon _r} = K = \frac{{{F_0}}}{{{F_m}}}\)
Where,
\(F_0\): The force acting between two charges in the air (Vacuum)
\(F_m\): Force acting between the same charges kept at the same distance in a medium
For vacuum (air), \(K = 1\)
For metals, \(K = \infty \)

Coulomb’s Law vs Universal law of Gravitation

The expression for electric force given by Coulomb is:
\({F_E} = \frac{{k{Q_1}{Q_2}}}{{{r^2}}}\)
The expression for gravitational force given by Newton is:
\({F_g} = \frac{{G{m_1}{m_2}}}{{{r^2}}}\)
From the above two expressions, we can see that both electric and gravitational forces have a lot in common. While the electric force depends on the product of charges, the gravitational force varies with the product of the masses. Both the forces vary inversely with the square of the distance between the bodies. Hence, both can be termed as ‘Inverse Square Law’. Both forces have a proportionality constant in their expressions. Both these forces are central and act along the line joining the two bodies.
The main difference between these two forces lies in their relative strengths. The electrostatic force between an electron and proton is several times greater than the gravitational force between them. Moreover, electric force is a short-range force, while gravitational force is a long-range force.

Limitations of Coulomb’s Law

1. The validity of Coulomb’s Law is determined by the number of molecules of solvent between the two charged bodies. If the average number of solvent molecules between the two interacting charged particles is large enough, we can only apply this law.
2. We can only apply Coulomb’s Law on stationary charges.
3. If the shape of the charged body is arbitrary, then we will not be able to determine the distance between the charges, and it becomes difficult to apply this law.

Summary

1. According to Coulomb’s Law, the electrostatic force acting between two charges, kept at a distance, varies directly with the product of two charges and inversely with the square of the distance between them. This force can be repulsive or attractive, depending on the polarities of the two charges.
2. The direction of Coulomb’s force is along the length of the line joining the centres of the two charges. It can be inwards or outwards, depending on the force of attraction or repulsion between the charges.
3. Coulomb’s law follows newton’s third law of motion.
4. Coulomb’s law of electrostatics holds for two or more point charges at rest.
5. The SI unit of Coulomb’s force is Newton.
6. The SI unit of permittivity is \(\frac{{{{\text{C}}^{\text{2}}}}}{{{\text{N}}{{\text{m}}^{\text{2}}}}}.\) It can also be expressed as farad per meter \(\left( {{\text{F/m}}} \right).\)

Frequently Asked Questions

Q.1. State coulomb’s law.
Ans: According to Coulomb’s law, the magnitude of the electrostatic force of attraction or repulsion between two point charges is directly proportional to the product of the two charges and inversely proportional to the square of the distance between them.

Q.2. The electrostatic force is maximum in air. Why?
Ans: The value of permittivity is least for air/vacuum. That is why Coulomb’s force is maximum in air.

Q.3. Why is Coulomb’s law called inverse square law?
Ans: According to Coulomb’s law, the electric force between two charges varies inversely with the square of the distance between the two charges. That is why it is also called the Inverse Square Law.

Q.4. Write the limitations of Coulomb’s law.
Ans: The formula can be applied with charges of regular and smooth shape, but it becomes complex when we are working with charges having irregular shapes. The formula is only valid for point charges and when the size of solvent molecules of the medium is significantly greater than the size of actual charges.

Q.5. What is the S.I. unit of relative permittivity?
Ans: Relative permittivity is a ratio of force acting between two charges in air and medium. It is a unitless quantity.

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