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Question: What is the force acting between two point charges?...

What is the force acting between two point charges?

Explanation

Solution

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∝abF \propto ab and F∝1r2F \propto \dfrac{1}{{{r^2}}}
F∝abr2F \propto \dfrac{{ab}}{{{r^2}}}
Or, F=kabr2F = k\dfrac{{ab}}{{{r^2}}}
Where k is constant of proportionality.
k=14πε0=9×109Nm2C−2k = \dfrac{1}{{4\pi {\varepsilon _0}}} = 9 \times {10^9}N{m^2}{C^{ - 2}} and
ε0=8.854×10−12N−1m−2C2{\varepsilon _0} = 8.854 \times {10^{ - 12}}{N^{ - 1}}{m^{ - 2}}{C^2}.
Thus force between two point charges is written as:
F=14πε0abr2F = \dfrac{1}{{4\pi {\varepsilon _0}}}\dfrac{{ab}}{{{r^2}}}.
It can also be represented in vector form as:
If force is on b due to a,
F⃗21=14πε0abr212r^21{\vec F_{21}} = \dfrac{1}{{4\pi {\varepsilon _0}}}\dfrac{{ab}}{{r_{21}^2}}{\hat r_{21}}

And 2) if force is on a due to b,
F⃗12=14πε0abr122r^12{\vec F_{12}} = \dfrac{1}{{4\pi {\varepsilon _0}}}\dfrac{{ab}}{{r_{12}^2}}{\hat r_{12}}
Where r^12 and r^21{\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 10−17m{10^{ - 17}}m to 107m{10^7}mand it is not affected by introducing any other charges.