It's a newbie question. How is the formula for force between two charges or gravitational force between two particles derived. Is it just intuition or is there any method used to derive these equations? For example I am not able to understand how is $r^2$ term of these formulas are taken.
Consider the gravitational force equation is not yet discovered. So what would be the steps someone need to reach in this formula?
Newtons law of gravitation is not just a speculation based on intuition. It is safe to say that $\frac{1}{r^2}$ law of gravitation is based on accurate experimental quantification by Kepler relating time period of orbit and semi major axis of the orbit goes by the rule Kepler's third law. I invite you to work out how the $\frac{1}{r^2}$ rule beautifully explains Kepler's relation.
It is known that planets move around the sun in elliptical orbits in Newton's time.
Now Coulomb discovered his $\frac{1}{r^2}$ law by conducting extensive and presise experiments. Coulomb used a torsion balance and measure the forces between charged objects leading to his law. Read more about the experiment here.
You cannot derive these type of physics equation .Newton's gravitational theory is just a theory. It is not accurate but approximation of complicated theories like general relativity. Same thing goes with electrostatic force. It was found Experimentally that these forces are proportional to masses or charges of the bodies. Furthermore newton using Kepler's law deduced that the force follows inverse square law. Finally by adding that constant of proportionality it completed the equation. The experimental results confirmed it so we accepted that theory. In 20th century a Einstein introduced a new theory called general relativity (for gravity) and it can explain things which newton's theory couldn't so we accepted the new theory.
