Today, it's the turn of the eleventh term in our Month of Equations series. Coming to the behavior of charged particles, we have discussed two equations in this regime till now, the Poisson's equation and the Gauss Law. But, I realised that till now we have not touched upon the most basic equation of Electrostatics, the Coulombs law. But as they say, It's never too late! So today, let's try to understand what Coulombs law, the law that governs all the electrostatic interactions in our Universe really says! According to Coulomb's Law,

the force between any two point charges is directly proportional to the product of their charges and is inversely proportional to the square of distance between them.

**Point charge:** In order to understand the above statement, first of all we need to understand the concept of a point charge. A point charge is a hypothetical charge located at a single point in space. Moreover, two charges can be considered point charges if the distance between them is much larger than their individual dimensions. It's just like if two table tennis balls are pinned at the ends of a string few kilometers long, then those balls can be considered as point charges.

### Mathematical expression for Coulomb's law:

Suppose, we have two point charges, having charge $latex {q_{1}}$ and $latex {q_{2}}$. If R is the distance between them, then the force inserted by one charge on the other is given as:

$latex { F \propto \frac{q_{1} q_{2}}{ R^{2}}}$

Inserting the value of proportionality constant $latex {k}$, where $latex {k = \frac{1}{4 \pi \epsilon_{0}}} $ , $latex {\epsilon_{0}}$ being the absolute permittivity of vacuum, we get

$latex {F = k \frac{q_{1} q_{2}}{ R^{2}}}$

On substituting the values of constants in S.I units, k comes out to be $latex {9 X 10^{9} Nm^{2}C^{-2}}$ If both the charges are same, ie. both are positive or both are negative, then the force will come with a positive sign, indicating repulsive nature. However, if the charges are opposite, ie. one is positive and the other is negative, then the expression for force comes with negative sign, indicating attractive force. This expression is analogous to the expression for gravitational force between two masses. Both the forces obey Inverse square law and both act along the line joining the two particles. Only the proportionality constant is different and the masses have been replaced with charges. Moreover, gravitational forces are always attractive , whereas Coulomb forces can be repulsive as well.

### Vector form of coulomb's law :

We know that force is a vector quantity. So, along with having a certain magnitude, it also has a particular direction. This means that the force exerted by first charge on the second one and vice verse won't be exactly same. There has to be a difference. This relation between the two forces is given by the vector form of Coulomb's law. The vector form of Coulomb's law is simply the scalar definition of the law with the direction given by the unit vector, r?21 , parallel with the line from charge q2 to charge q1 and vice versa for r?12.

As r?21 = - r?12. This shows that force exerted by first charge on the second and by second charge on first one are exactly equal in magnitude and opposite in direction. ie. $latex {F_{12} = -F_{21} }$. This confirms Newton's third law.

### Author's message:

Coulomb's law is one of the most basic laws which we come across in school while beginning our journey in Electrostatics. It highlights the inverse square dependence of electric forces and also forms the basis for some heavier derivations like Gauss's law.No doubt, It is governing all the charged particles in our Universe. Still most people fail the understand and realise the beauty and meaning of it. So, in this article, I aimed to highlight the richness and beauty of the simple Coulombs law. I hope you had a smooth read!

*Previous in serie**s** : Meaning of Gauss' Law in Electrodynamics*

*Editor* at **The Secrets of the Universe**, I am a science student pursuing Master’s in Physics from India. I love to write about Cosmology, Condensed Matter Physics and Quantum Mechanics.

Julius AnggotThe quantity of electrostatic force between stationary charges is always described by Coulomb's law. Thank you very much!