What is Coulomb’s Law?

Coulomb's law states that the attraction or repulsion force between two charged things is proportional to the product of their charges and inversely proportional to the square of their distance. It has an effect along the line that connects the two-point charges.

What is the principle of Coulomb’s Law?

Coulomb’s law is used to calculate the amount of force between the two stationary electrically charged particles. The electric force between charged bodies at rest is called electrostatic force or Coulomb’s force.

Coulomb’s law states that the magnitude of the force of attraction or repulsion between two point charges is directly proportional to the product of charges and inversely proportional to the square of the distance between them. Two positively or two negatively charged particles repel each other whereas, two opposite charges attract each other.

Coulomb’s Law Formula

In scalar form, the force $F$ between the two-point charges $\left(q_1\right)$, and $\left(q_2\right)$ is directly proportional to the product of charges $\left(q_1{q}_2\right)$ and inversely proportional to the square of the distance ($r^2$) between them. The derived equation for Coulomb’s electric force is as shown

$$\begin{gathered} F\propto \frac{q_1{q}_2}{r^2} \\ F=k\frac{q_1{q}_2}{r^2} \\ F=\frac{1}{4{\mathrm{\pi \epsilon }}_0}\frac{q_1{q}_2}{r^2} \end{gathered}$$

Here, ${\epsilon }_0$ is the permittivity of the free space or vacuum. The value of $k=9\times {10}^9 {\mathrm{Nm}}^2{\mathrm{C}}^{-2}$.

Coulomb’s law in vector form

Let both the charges be negative. As a result, the electrostatic force between the two particles is repulsive because of same charges repel each other. For two given charges, force ${\stackrel{\rightharpoonup }{F}}_{12}$ is exerted on the charge $q_1$ due to charge $q_2$.  Similarly, ${\stackrel{\rightharpoonup }{F}}_{21}$ is the force exerted on charge $q_2$ due to $q_1$.

The mathematical equation relation for the electrostatic force is depicted below.

${\stackrel{\rightharpoonup }{F}}_{12}=\frac{1}{4{\mathrm{\pi \epsilon }}_0}\frac{q_1{q}_2}{r_{12}^2}{\hat{r}}_{12}$

Here, $r_{12}$ is the separation distance between the two charged particles, and ${\hat{r}}_{12}$ is the unit vector in the direction of the force ${\stackrel{\rightharpoonup }{F}}_{12}$.

Coulomb’s law follows Newton’s third law of motion. As a result, the force ${\stackrel{\rightharpoonup }{F}}_{12}$ exerted on the charge $q_1$ due to charge $q_2$ is equal in magnitude but opposite in direction to ${\stackrel{\rightharpoonup }{F}}_{21}$ which is the force exerted on charge $q_2$ due to $q_1$. It can be deduced mathematically with the following equation.

${\stackrel{\rightharpoonup }{F}}_{12}=-{\stackrel{\rightharpoonup }{F}}_{21}$

Conditions for the stability of the Coulomb force

Force $\left(F_A\right)$ on the charge located at point A increases in amplitude while the force $\left(F_B\right)$ on the charge located at point B lowers when the charge $q$ is slightly shifted toward A. Because q's net force is now directed toward A, it will not return to its former position. As a result, the equilibrium for axial displacement is unstable.

However, if the charge $\left(q\right)$ is displaced perpendicular to the line joining A and B, the force $\left(F_A\right)$ on the charge located at point A and the force $\left(F_B\right)$ on the charge located at point B brings the charge to its original position. As a result, for perpendicular displacement, the equilibrium is stable.

Coulomb’s Constant

Coulomb's constant is a proportionality factor found in Coulomb's law and other electric formulas or equations. It is denoted by $k_e$. It is also known as an electric force constant or electrostatic constant. The standard international units of all the parameters in the Coulomb law equation are used to derive the unit of the Coulomb constant. The force SI unit is newtons, charges in coulombs and distance in metres. The SI unit of the electrostatic constant is depicted below:

$$\begin{gathered} F=k_e\frac{q_1{q}_2}{r^2} \\ N=k_e\frac{C\left(C\right)}{{\mathrm{m}}^2} \\ {k}_e={\mathrm{Nm}}^2{\mathrm{C}}^{-2} \end{gathered}$$

The numeric value of the electrostatic constant is $k_e=8.98\times {10}^9 {\mathrm{Nm}}^2{\mathrm{C}}^{-2}$.

Further, electrostatic can be expanded in terms of permittivity. The electric constant is inversely proportional to the electric permittivity in the vacuum. The SI unit of electric permittivity from Coulomb’s law formula is depicted below:

$$\begin{gathered} k_e=\frac{1}{4{\mathrm{\pi \epsilon }}_0} \\ {\mathrm{\epsilon }}_0=\frac{1}{4{\mathrm{\pi k}}_{\mathrm{e}}} \\ {\mathrm{\epsilon }}_0=\frac{1}{{\mathrm{Nm}}^2{\mathrm{C}}^{-2}} \\ {\mathrm{\epsilon }}_0={\mathrm{N}}^{-1}{\mathrm{m}}^{-2}{\mathrm{C}}^2 \end{gathered}$$

The constant has different, dimensionless Gaussian and Lorentz–Heaviside units, both CGS unit systems.

In electrostatic units or Gaussian units, the unit charge is elaborated in such a way that the Coulomb constant disappears because its value becomes one, so dimensionless.

$k_{e\left(Gaussian units\right)}=1$

The Lorentz-Heaviside units are known as rationalized units. The Coulomb constant associated with Lorentz-Heaviside units is dimensionless and is equal to one by $4\mathrm{\pi }$.

For microscopic problems like the electrodynamics of single electrically charged particles, Gaussian units are better suitable. For practical, large-scale phenomena like engineering applications, SI units are more convenient.

Meaning of one coulomb charge

One Coulomb charge is that charge that repels an equal charge of the same sign either both positive or both negative with a force of $9\times {10}^9 \mathrm{N}$ two-point charges being one meter apart in a vacuum. Coulomb forces are conservative and it is an internal force.

Limitations of Coulomb's law

Coulomb's inverse square law must satisfy three conditions to be valid. These are as follows:

• The charges must be distributed in a spherically symmetric manner. Specifically, the charges should be point-sized, or the charges should be charged metal spheres.
• The charges, regardless of whether negative or positive, must not overlap; that is, they must be separated by some distance. In other words, we can say that the point charges should be distinct.
• The charges must be stationary concerning each other.

The electrostatic approximation can be elaborated as charges should be stationary concerning each other. When movement occurs, Einstein's theory of relativity must be considered, resulting in the introduction of a new element that affects the force produced on the two objects. Magnetic fields describe this extra half of the force, known as the magnetic force. When the charges travel slowly around each other, the magnetic force is negligible, and Coulomb's law can still be considered roughly right. However, when the charges move faster about each other, the entire electrodynamics laws (including the magnetic force) must be considered.

Uses of Coulomb’s law

The real-life applications of Coulomb law are as follows:

• Xerox photostat is an application of Coulomb's law. The electrostatic process is responsible for copying. It employs a selenium-coated metal drum because selenium has an intriguing property: in the dark, it acts as an insulator, but when exposed to light, it acts as a conductor.

• A high voltage electrostatic charge is applied to both the object to be coated and the sprayer mechanism in the powder coating process.
• Laser printing

Common Mistakes

Remember that the electrostatic force can be calculated on any charge only when both the charges are point-sized and stationary.

Context and Applications

In each of the expert exams for undergraduate and graduate publications, this topic is huge and is mainly used in the following context:

• Bachelor of Technology in the Electrical and Electronics Department
• Bachelor of Science in Physics
• Master of Science in Physics

Related Concepts

• Electric field
• Electric charge
• Gauss law

Practice Problems

Q1. If one particle carries a positive charge and the other carries a negative charge, then the force between them is __________.

1. repulsive
2. attractive
3. repulsive and attractive
4. cannot be determined

Correct option: (b)

Explanation: The force between the two opposite charges is attractive. As a result, the force between the one particle being negatively charged and the other being positively charged is attractive.

Q2. How is electrostatic force between the two charged particles dependent on the distance between them?

1. Directly proportional
2. Linear
3. Inversely square
4. Exponentially dependent

Correct option: (c)

Explanation: As Coulomb law, the electrostatic force between the two charged particles is inversely proportional to the product of charges and inversely proportional to the square of the distance between them.

Q3. The SI unit of force and charge, respectively, is __________.

1. newton and coulomb
2. millinewton and kilocoulomb
3. metre and ampere
4. volt and ampere

Correct option: (a)

Explanation: The SI unit of force is Newton, and the SI unit of charge is the coulomb.

Q4. The value of electric constant k in coulomb’s law is _________.

1. $8.85\times {10}^{-12} {\mathrm{C}}^2{\mathrm{N}}^2{\mathrm{m}}^{-2}$
2. $9\times {10}^9 {\mathrm{N}}^2{\mathrm{m}}^2{\mathrm{C}}^{-2}$
3. $9\times {10}^9 {\mathrm{N}}^2{\mathrm{m}}^{-2}{\mathrm{C}}^{-2}$
4. None of the bove

Correct option: (b)

Explanation: The numeric value of the electric constant is $9\times {10}^9$, and its SI unit is ${\mathrm{N}}^2{\mathrm{m}}^2{\mathrm{C}}^{-2}$. As a result, $9\times {10}^9 {\mathrm{N}}^2{\mathrm{m}}^2{\mathrm{C}}^{-2}$ in Coulomb’s law.

Q5. If the distance between the two charged bodies is halved, the force between them becomes __________.

1. one fourth
2. double
3. four times
4. None of these

Correct option: (c)

Explanation: According to Coulomb law, the force between the two charges is directly proportional to the product the changes and inversely proportional to the square of the distance between them. If the distance is halved, the force between the two charges becomes four times.