# Coulomb’s Law of electrostatic force |formula, vector form

Earlier we have discussed the electric charge and its properties. In other articles we have learned that an electric charge can produce electric field and magnetic field (if the charge is moving) around it. Inside this field region it can interact with other charges. This interaction is nothing but the force between the charges. This force of the electric charges is known as the Electrostatic Force or the electric force. Coulomb’s law of electrostatics gives the equation or the expression for electrostatic force acting between the charges. In this article, we are going to discuss the Coulomb’s Law of electrostatic force.

• Fundamental Law of electrostatics
• Coulomb’s law of electrostatic force
• Formula for Coulomb’s Law
• Vector form of Coulomb’s Law
• Properties of Electrostatic force
• Comparison between Coulomb’s electrostatic force and Gravitational force
• Derivation of Coulomb’s Law from Gauss’s Law
• Application of Coulomb’s Law
• Limitations of Coulomb’s law

## Fundamental Law of electrostatics

We already know that electric charges exert forces on each other. The Fundamental law of electrostatics states that the electric charges of same sign repel each other and the electric charges of opposite sign attract each other. That means two positive charges or two negative charges repel each other. But if we place a positive charge near a negative charge, then they will attract each other.

## Coulomb’s law of electrostatic force

Coulomb’s law of electrostatics gives an expression for the electric force between the electric charges. This law states that the electrostatic force between two point charges separated by a distance is directly proportional to the multiplication between the magnitude of the charges and inversely proportional to the square of the distance between the charges.

Mathematical equation of Coulomb’s law is given below.

## Formula for Coulomb’s lawof electrostatics

Let two point charges +Q and +q are placed at a distance of separation r. Then according to Coulomb’s law, the electric force (F) acting between the charges is,

$\small {\color{Blue} F\propto Qq}$ and, $\small {\color{Blue}F\propto \frac{1}{r^{2}}}$

Then we get, $\small {\color{Blue} F\propto \frac{Qq}{r^{2}}}$

or, ${\color{Blue}F=\frac{kQq}{r^{2}}}$ ………………….. (1)

Where, k is the Coulomb’s constant. It varies medium to medium.

In CGS unit, the value of k is 1 and in SI unit, k has an expression as ${\color{Blue} k=\frac{1}{4\pi \epsilon _{0}}}$ in air medium or free space which have the value of 9×109. Here, $\small {\color{Blue} \epsilon _{0}}$ is the permittivity of free space.

So, the equation for Coulomb’s law in air medium is,

$\small {\color{Blue} F=\frac{1}{4\pi \epsilon _{0}}\frac{Qq}{r^{2}}}$ …………… (2) in SI unit.

and, $\small {\color{Blue} F=\frac{Qq}{r^{2}}}$ ………………. (3) in CGS unit.

For the other medium, one should use the permittivity of that medium in place of the permittivity of free space.

## Vector form of Coulomb’s Law

In SI system, the magnitude of electrostatic force is given by the equation-(2). Now, the force is repulsive for two positive charges +Q and +q. So, the force on q will act along the outward direction from q. We denote the unit vector by ${\color{Blue} \widehat{r}}$ along the outward direction from q.

Then the vector form of Coulomb’s law is, $\small {\color{Blue} \overrightarrow{F}=\frac{1}{4\pi \epsilon _{0}}\frac{Qq}{r^{2}}\widehat{r}}$ …….. (4)

Again, the position vector is, $\small {\color{Blue} \overrightarrow{r}=r\widehat{r}}$. Then, $\small {\color{Blue} \widehat{r}=\frac{\overrightarrow{r}}{r}}$……….. (5)

Now, from equation-(4) and equation-(5) one can get, $\small {\color{Blue} \overrightarrow{F}=\frac{1}{4\pi \epsilon _{0}}\frac{Qq}{r^{3}}\overrightarrow{r}}$ ……… (6).

Equation-(6) is the another form of the vector form of Coulomb’s law.

## Properties of Coulomb’s electrostatic force

1. Electrostatic force is a Conservative force.
2. This force acts between the electric charges.
3. Electrostatic force may be attractive or repulsive.
4. This force obeys inverse square law of the distance between the charges. Magnitude of this force decreases with increase in distance between the charges.
5. Coulomb’s Electrostatic force depends on the medium. The magnitude of electrostatic force for the same two charges and same distance is different for different medium.
6. Electrostatic force is not applicable for the charges placed at a distance smaller than the nuclear distance.

## Comparison between Coulomb’s electrostatic force and Gravitational force

There are some similarities and also some differences between electrostatic force and Gravitational force. Here we are going to discuss those.

### Similarities between Electrostatic force and Gravitational force

1. Electrostatic force and the Gravitational force are the conservative forces.
2. Both of the Electrostatic and Gravitational force obey inverse square law of the distance.

### Differences between Electrostatic force and Gravitational force

1. Electrostatic force may be attractive or repulsive. But Gravitational force is always attractive in nature.
2. Coulomb’s Electrostatic force between the charges depends on the medium where the charges are placed. But the Gravitational force is independent of the medium.
3. Electrostatic force is much stronger than the Gravitational force.

## Derivation of Coulomb’s Law of electrostatic force from Gauss’s Law

According to the Gauss’s law of electrostatics, the electric flux passing through a spherical surface of charge Q and radius r is, $\small {\color{Blue} \int \overrightarrow{E}.\overrightarrow{dS}=\frac{Q}{\epsilon _{0}}}$.

Then the electric field on the surface of the sphere due the Q charge is, $\small {\color{Blue} E=\frac{Q}{4\pi \epsilon _{0}r^{2}}}$.

Now, we place another charge q on the surface of the sphere. Then the electrostatic force on the charge q will be,

F=qE or, $\small {\color{Blue} F=\frac{Qq}{4\pi \epsilon _{0}r^{2}}}$.

This is nothing but the Coulomb’s Law. Hence, Coulomb’s law is derived from the Gauss’s law.

## Application of Coulomb’s law

The only application of Coulomb’s law is to find the electrostatic force between the electric charges. One can find the value of electric field at any point by observing the Coulomb’s force at that point.

## Limitations of Coulomb’s law

There are some limitations of Coulomb’s law –

1. Coulomb’s law is applicable only for the point charges. To apply this law for a large body its total charge is assumed to be located at a single point (at Centre of charge) so that it can be treated as a point charge.
2. This law is valid for the charges at rest with respect to the observer.
3. Coulomb’s law is valid when the distance between the charges is greater than the nuclear distance or nuclear diameter (10-15m).

This is all from this article. If you have any doubt on this topic you can ask me in the comment section.

Thank you!

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