How does one come up with the Coulomb's law?

Question

My teacher mentioned that field line density = no. of lines / area and the total area of a sphere is $4\pi r^2$ and so an electric force is inversely proportional to $r^2$. Actually, why can the total area of the sphere be applied to this case and is this true? How does one come up with the Coulomb's law?

Answer 1

Technically speaking, Coulomb's law "came first". It is the result of multiple experiments conducted on charges, which showed that the force came from an inverse-square law.

Your teacher here is actually assuming Gauss' law to be a fundamental and carrying forward. Gauss' law basically says that field line density is directly proportional to charge contained in the sphere (since "number of lines" is directly proportional), and that field line density itself is just another word for electric field intensity. In other words: $$E\propto\frac{\text{number of lines}}{\pi r^2}\propto\frac{q}{r^2}$$

The formal expression of Gauss' law is that flux of the field through a given surface $\partial W$ is proportional to the charge enclosed:

$$\iint_{\partial W}\mathbf E\cdot d\mathbf S=\frac{q_{enc}}{\epsilon_0}$$

If we take a sphere around a charge, the electric field is uniform and perpendicular at all points on the sphere (by symmetry), so the left hand side can be rewritten as $$\iint_{\partial W}\mathbf E\cdot d\mathbf S= E\iint_{\partial W}dS=E\times4\pi r^2$$ and by rearranging, we get Coulomb's law.

Don't let this fool you, though -- Though Gauss' law is many a times considered fundamental (you can always choose what you want to consider as "fundamental", as long as the system is consistent), it is historically derived from Coulomb's law.