Question

What is the force acting between two point charges?

Answer 1

Hint: There was an experiment conducted by Coulomb, in 1785. He gave a statement to explain what the force acting between two point charges is. A point charge is a charge assumed to be at a geometric point. It has no dimensions. Here we will start by assuming 2 charges separated by a distance r and then derive COULOMB'S LAW.

Complete answer:
We will learn about a very interesting topic now called COULOMB’S LAW.
This law states that the force of attraction or repulsion between 2 point charges is directly proportional to the product of the magnitudes of charges and inversely proportional to the square of distance between them, and acts along the line joining the 2 charges.
 Let’s derive this,
Let there be 2 point charges a and b separated by distance r, look at the following image:

Hence applying the law we get: $$F\propto ab$$ and $$F\propto {\displaystyle \frac{1}{r^2}}$$
$$F\propto {\displaystyle \frac{ab}{r^2}}$$
Or, $$F=k{\displaystyle \frac{ab}{r^2}}$$
Where k is constant of proportionality.
$$k={\displaystyle \frac{1}{4\pi {\epsilon }_0}}=9\times {10}^{9}N{m}^2{C}^{-2}$$ and
$${\epsilon }_0=8.854\times {10}^{-12}{N}^{-1}{m}^{-2}{C}^2$$.
Thus force between two point charges is written as:
$$F={\displaystyle \frac{1}{4\pi {\epsilon }_0}}{\displaystyle \frac{ab}{r^2}}$$.
It can also be represented in vector form as:
If force is on b due to a,
$${\overrightarrow{F}}_{21}={\displaystyle \frac{1}{4\pi {\epsilon }_0}}{\displaystyle \frac{ab}{r_{21}^2}}{\hat{r}}_{21}$$


And 2) if force is on a due to b,
$${\overrightarrow{F}}_{12}={\displaystyle \frac{1}{4\pi {\epsilon }_0}}{\displaystyle \frac{ab}{r_{12}^2}}{\hat{r}}_{12}$$
Where $${\hat{r}}_{12}and{\hat{r}}_{21}$$ are the unit vectors from a to b and b to a respectively.

Note: The unit vectors are mentioned to tell that the force can be from object a to b or from object b to a. It is always better to mention the vector quantities because it specifies direction along with magnitude. Also remember that Coulomb’s law is valid in a vast range $${10}^{-17}m$$ to $${10}^{7}m$$and it is not affected by introducing any other charges.