The force between charged bodies is no contact force. It exists over a length, and all electrical interaction has a force embedded in it. The charges and distance between the charged bodies are the factors that determine the power and influence of the force. The same force exists, whether it’s a plastic comb attracting paper pieces or two electrons repelling each other.

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## What Is Coulomb’s Law?

According to **Coulomb’s law**, the force of attraction or repulsion between two charged bodies is directly proportional to the product of their charges and inversely proportional to the square of the distance between them. It acts along the line joining the two charges considered to be point charges.

### Table of Content

- Coulomb’s Law Formula
- Coulomb’s Law in Vector Form
- What Is One Coulomb of Charge
- Key Points
- Limitations
- Relative Permittivity
- Applications
- Problems

**Coulomb’s Law Formula**

In short, F โ q_{1}q_{2}/d^{2}

**Where,**

**ฮต is absolute permittivity**,**K**or**ฮต**is the_{r}**relative permittivity**or**specific inductive capacityย****ฮต**is the_{0}**permittivity of free space**.- K or
**ฮต**is also called a dielectric constant of the medium in which the two charges are placed._{r}

### History of Coulomb’s Law

In 1785, French physicist Charles Augustin de Coulomb coined a tangible relationship in mathematical form between two bodies that have been electrically charged. He published an equation for the force causing the bodies to attract or repel each other, which is known as Coulomb’s law or **Coulomb’s inverse-square law**.

## Coulomb’s Law in Vector Form

Here, F_{12} is the force exerted by q_{1} on q_{2,} and F_{21} is the force exerted by q_{2} on q_{1}.

Coulombโs law holds for stationary charges only, which are point sized. This law obeys Newtonโs third law

Force on a charged particle due to a number of point charges is the resultant of forces due to individual point charges, i.e.,

## What Is 1 Coulomb of Charge?

A Coulomb is a charge which repels an equal charge of the same sign with a force of 9ร10^{9} N when the charges are one metre apart in a vacuum. Coulomb force is the conservative mutual and internal force.

The value ofย ฮต_{o}ย is 8.86 ร 10^{-12}ย C^{2}/Nm^{2}ย (or) 8.86 ร 10^{-12}ย Fm^{โ1}

**Note:**ย Coulomb force is true only forย static charges.

## Coulomb’s Law – Conditions for Stability

If q is slightly displaced towards A, F_{A} increases in magnitude while F_{B} decreases in magnitude. Now, the net force on q is toward A, so it will not return to its original position. So for axial displacement, the equilibrium is unstable.

If q is displaced perpendicular to AB, the force F_{A} and F_{B} bring the charge to its original position. So, for perpendicular displacement, the equilibrium is stable.

## Key Points on Coulomb’s Law

**1.** If the force between two charges in two different media is the same for different separations,

**2.ย ** Kr^{2} = constant or K_{1}r_{1}^{2} = K_{2}r_{2}^{2ย }

**3.** If the force between two charges separated by a distance โr_{0}โ in a vacuum is the same as the force between the same charges separated by a distance โrโ in a medium, then from Coulomb’s law, Kr^{2} = r_{0}^{2}

**4.** Two identical conductors having charges q_{1} and q_{2} are put to contact and then separated, after which each will have a charge equal to (q_{1} + q_{2})/2. If the charges are q_{1} and โq_{2}, each will have a charge equal to (q_{1} – q_{2})/2.

**5.** Two spherical conductors having charges q_{1} and q_{2} and radii r_{1} and r_{2} are put to contact, and then separated the charges of the conductors after contact is

q_{1} = [r_{1}/(r_{1} + r_{2})] (q_{1} + q_{2}) and q_{2} = [r_{2}/(r_{1} + r_{2})] (q_{1} + q_{2})

**6.**ย If the force of attraction or repulsion between two identical conductors having charges q_{1} and q_{2} when separated by a distance d is F. Also, if they are put to contact and then separated by the same distance, the new force between them is:

**7.ย **If charges are q_{1} and -q_{2,} then F = F(q_{1} + q_{2})^{2} / 4q_{1}q_{2}

**8.** Between two electrons separated by a certain distance: Electrical force/Gravitational force = 10^{42}

**9.** Between two protons separated by a certain distance: Electrical force/Gravitational force = 10^{36}

**10.** Between a proton and an electron separated by a certain distance: Electrical force/Gravitational force = 10^{39}

**11.** The relationship between the velocity of light, the permeability of free space and the permittivity of free space is given by the expression c = 1 / โ (ฮผ_{o}ฮต_{o} )

**12.** If Coulomb’s law is applied to two identical balls of mass m are hung by silk thread of length โlโ from the same hook and carry similar charges q, then,

- The distance between balls \(\begin{array}{l}=[\frac{q^{2}2l }{4\pi \epsilon _{o}mg}]^{\frac{1}{3}}\end{array} \)
- The tension in the thread \(\begin{array}{l}=\sqrt{f^2+(mg)^2}\end{array} \)
- If the total system is kept in space, then the angle between threads is 180ยฐ, and tension in a thread is given by

- A charge Q is divided into q and (Q โ q). Then, the electrostatic force between them is maximum when

## Limitations of Coulomb’s Law

- The law is applicable only for the point charges at rest.
- Coulomb’s law can only be applied in those cases where the inverse square law is obeyed.
- It is difficult to implement Coulombโs law where charges are in arbitrary shape because, in such cases, we cannot determine the distance between the charges.
- The law can’t be used directly to calculate the charge on big planets.

### Relative Permittivity of a Material

- For air K = 1
- For metals, K = infinity

The force between 2 charges depends on the nature of the intervening medium, whereas gravitational force is independent of the intervening medium.

For air or vacuum,

The value of 1/4ฯฮต_{0}ย is equal to 9 ร 10^{9} Nm^{2}/C^{2}.

#### โ Related Topics

## Application of Coulomb’s Law

- To calculate the distance and force between the two charges
- The electric field can be calculated using Coulomb’s law

Where, E = Strength of the electric field

F = Electrostatic force

Q_{T }= Test charge in Coulombs

- To calculate the force on one point due to the presence of several points (Theorem of superposition)

## Problems on Coulomb’s Law

**Problem 1: Charges of magnitude 100 microcoulomb each are located in a vacuum at the corners A, B and C of an equilateral triangle measuring 4 meters on each side. If the charge at A and C are positive and the charge at B negative, what is the magnitude and direction of the total force on the charge at C?**

**Sol.**

The situation is shown in the figure below. Let us consider the forces acting on C due to A and B.

Now, from Coulomb’s law, the force of repulsion on C due to A, i.e., FCA in the direction of AC, is given by

The force of attraction on C due to B, i.e., F_{CB} in direction CB, is given by

Thus, the two forces are equal in magnitude. The angle between them is 120ยบ. The resultant force F is given by

This force is parallel to AB.

**Problem 2:** **The negative point charges of unit magnitude and a positive point charge q are placed along the straight line. At what position and for what value of q will the system be in equilibrium? Check whether it is stable, unstable or neutral equilibrium.**

**Sol.**

The two negative charges – A and B – of unit magnitude are shown in the figure below. Let the positive charge q be at a distance r_{A} from A and at a distance r_{B} from B.

Now, from Coulomb’s law, force on q due to A

Force on q due to B

These two forces acting on q are opposite and collinear. For the equilibrium of q, the two forces must also be equal, i.e.,

|F_{qA}| = |F_{qB}|

or

Hence, rA = rB

So, for the equilibrium of q, it must be equidistant from A & B, i.e., in the middle of AB.

Now for the equilibrium of the system, A and B must be in equilibrium.

For the equilibrium of A,

Force on A by qย

Force on A by B

The two forces are opposite and collinear. For equilibrium, the forces must be equal, opposite and collinear. Hence,

or q = 1/4 in the magnitude of either charge

It can also be shown that for the equilibrium of B, the magnitude of q must be 1/4 of the magnitude of either charge.

**Problem 3: A positive charge of 6ร10 ^{-6ย }C is 0.040m from the second positive charge of 4ร10^{-6ย }C. Calculate the force between the charges.**

**Given **

**q _{1 =ย } 6ร10^{-6ย }C**

**q _{2ย }=ย 4ร10^{-6ย }C**

**r = 0.040 m**

**Sol.**

F_{eย }= 134.85 N

**Problem 4:ย Two-point charges, q _{1ย }=ย +9 ฮผC and q_{2ย }= 4 ฮผC, are separated by a distance r = 12 cm. What is the magnitude of theย electric force?ย **

**givenย **

**k = 8.988 x 10 ^{9}ย Nm^{2}C^{โ2
}**

**q _{1ย }= 9ย ร10^{-6} C**

**q _{2} = 4 ร10^{-6} C**

** r = 12cm = 0.12 m**

**Sol:**

F_{eย }= 22.475 N

### Coulomb’s Law and Superposition Principle – Electrostatics – JEE Main 2023

### Coulomb’s Law – Concepts and Questions

## Frequently Asked Questions on Coulombโs Law

### State Coulombโs inverse-square law in electrostatics.

The electrostatic force of attraction or repulsion between two stationary point charges is directly proportional to the product of the magnitudes of the charges and inversely proportional to the square of the distance between them.

### What are the limitations of Coulombโs law?

Coulombโs law holds good for stationary point charges.

Coulombโs law is not universal, as it depends on the properties of the intervening medium.

### Is the electrostatic force between two point charges a central force?

Yes. The electrostatic force between two point charges always acts along the line joining the two charges. Hence, it is a central force.

### What is one Coulomb of charge?

One Coulomb of charge is that charge which, when placed at rest in a vacuum at a distance of one metre from an equal and similar stationary charge, repels it and is repelled by it with a force of 9 x 10^{9} Newton.

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