The derivation of Coulomb's Law of electrostatics using Maxwell's equations:

Coulomb's Law describes the electrostatic force between two electrically charged particles. It states that the force between two point charges is directly proportional to the product of their magnitudes and inversely proportional to the square of the distance between them. Maxwell's equations, on the other hand, are a set of fundamental equations that describe how electric and magnetic fields interact. By using Maxwell's equations, we can derive Coulomb's Law.

Maxwell's equations:
Maxwell's equations are a set of four fundamental equations that describe the behavior of electric and magnetic fields:

1. Gauss's Law for Electric Fields: The electric flux through a closed surface is proportional to the net charge enclosed by the surface.
2. Gauss's Law for Magnetic Fields: The magnetic flux through a closed surface is zero.
3. Faraday's Law of Electromagnetic Induction: The electromotive force (EMF) induced in a closed loop is equal to the negative rate of change of magnetic flux through the loop.
4. Ampere's Law with Maxwell's Addition: The circulation of the magnetic field around a closed loop is equal to the sum of the current passing through the loop and the displacement current.

Derivation:
To derive Coulomb's Law using Maxwell's equations, we consider a point charge q1 at the origin and a test charge q2 at a distance r from it.

1. Gauss's Law for Electric Fields: Applying Gauss's Law, we consider a Gaussian surface enclosing the charge q1. The electric flux through the Gaussian surface is given by ΦE = q1/ε0, where ε0 is the permittivity of free space.
2. Gauss's Law for Magnetic Fields: Since there are no magnetic fields involved in this scenario, this law does not contribute to the derivation.
3. Faraday's Law of Electromagnetic Induction: Since there are no changing magnetic fields involved, this law does not contribute to the derivation.
4. Ampere's Law with Maxwell's Addition: Applying Ampere's Law, we consider a circular loop of radius r centered at the origin. The circulation of the magnetic field around this loop is given by ∮B⋅dl = μ0I, where μ0 is the permeability of free space and I is the current passing through the loop. In this case, there is no current passing through the loop, so the circulation is zero.

From the above analysis, we can conclude that the electric field due to a point charge is conservative. Therefore, we can define an electric potential function V such that the electric field is the negative gradient of the potential, E = -∇V.

Using this definition and the fact that the electric field is conservative, we can derive Coulomb's Law. Considering the electric field due to q1 at the position of q2, we have E = k(q1/r^2) where k = 1/(4πε0). Since the electric field is the negative gradient of the potential, we have ∇V = -k(q1/r^2).

Integrating both sides of this equation with respect to r, we get V = -k(q1/r) + C, where C is an integration constant. At infinity, the potential approaches zero, so we have V = -k(q1/r).

Now