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Maxwell's equations are a set of partial differential equations that, together with the Lorentz force law, form the foundation of classical electromagnetism, classical optics, and electric circuits. My question is not on its thermodynamic relation but I want someone to please help me state the various equation and the theorems they are being derived from.

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    $\begingroup$ Maxwell’s equations are physical laws. They do not follow from just mathematical theorems. $\endgroup$ – G. Smith Apr 3 at 2:26
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    $\begingroup$ You should make your question more precise: what is it that you would like to know? Well-posedness and uniqueness of solutions? Are you talking about the in-vacuo equations or do you include those for matter? What are the constitutive relations then, and what assumptions do you make on those? Moreover, the Lorentz force law, for example, is not part of Maxwell's equations. $\endgroup$ – Max Lein Apr 3 at 2:37
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Basically I see 2 ways to answer your question:

Empirical Derivation

Gauss's Law:

$$\nabla \cdot \mathbf{E} = \frac{\rho}{\epsilon_0}.$$

In words, this law states that the sources of electric fields are charges. The derivation comes from looking at the total electric flux in through a closed surface S, that should be equal to the total charge in the volume:

$$\Phi_E = \oint_S \mathbf{E} \cdot \text{d} \mathbf{A} = \frac{Q}{\epsilon_0}.$$

By using the Gauss divergence theorem and expressing the total charge as the volume integral of the charge density $Q = \int_V \rho \text{d}V$, one arrives at the above equation.

divBzero:

$$\nabla \cdot \mathbf{B} = 0.$$

Similar to the above one, this states that there is no source for magnetic fields, i.e. no magnetic monopoles exist. This could, of course, be reformulated with magnetic charge density if it turns out that magnetic monopoles do exist, but by now there was absolutely no evidence.

Faraday's Law:

$$\nabla \times \mathbf{E} = -\frac{\partial \mathbf{B}}{\partial t}.$$

This gives a link between time varying magnetic fields and the change of the electric field due to that. It can be derived (or has a connection) to the induction equation by Faraday. This states, that the negative time derivative of the magnetic flux around a closed path is the electromotive force around this path:

$$- \frac{\text{d}\Phi_B}{\text{d}t} = \text{EMF}.$$

Using the definition of the magnetic flux $\Phi_B = \int_{S(t)} \mathbf{B}(t) \cdot \text{d}\mathbf{A}$ (similar to electric flux above, but time dependent) and taking the EMF as the integral of the Lorentz-force along the closed path

$$\text{EMF}=\oint_{\partial S(t)}(\mathbf{E} + \mathbf{v} \times \mathbf{B}) \cdot \text{d}\mathbf{l},$$

one can derive Faraday's Law from this (It is not so easy, you can have a look at this Wikipedia-entry).

Amperé's Law:

$$\nabla \times \mathbf{B} = \mu_0 \mathbf{J} + \mu_0 \epsilon_0 \frac{\partial \mathbf{E}}{\partial t}.$$

First, only including the first term on the right side, this states that the change in magnetic fields is caused by currents. Including the second one, there is an additional term that gives rise to a change in magnetic field: the displacement current which is due to time varying electric fields.

The initial form (including only the first term on the right) was derived by Maxwell himself in analogy to hydrodynamics. After realizing, it did no fit with the experiments (expecially with predictions on condensators) he had to modify it, introducing the displacement current.

Derivation From Action Principle

I'm not sure if you know relativistic electrodynamics (especially you need the Minkowski formalism for this), but it is a beautiful way to derive all Maxwell equations from a given Lagrange density (which was found out back then by trial). This is analogous from classical mechanics, where you can derive the equations of motions from a given Lagrange density.

You know, that the magnetic and electric field can be written in terms of potentials:

$$\mathbf{E} = - \nabla \phi - \frac{\partial \mathbf{A}}{\partial t},\,\,\, \mathbf{B} = \nabla \times \mathbf{A}.$$

Here, $\phi$ is the usual scalar potential and $\mathbf{A}$ the usual vector potential. One can write them together into a 4-vector, resulting in the so called 4-potential:

$$A_\mu(x) = (\phi(x)/c, \mathbf{A}(x)),$$

Greek indices $\mu, \nu, ... = 1...4$. Similar for the charge and the current density. The 4-current is:

$$j_\mu(x) = (c\rho(x), \mathbf{j}(x)).$$

Now, the relation between the fields and the potentials can be written in compact form. This results in the field strength tensor:

$$F_{\mu\nu} = \partial_\mu A_\nu - \partial_\nu A_\mu.$$

Now, one can use the Lagrange density of electrodynamics:

$$\mathcal{L} = - \frac{1}{4} F_{\mu\nu}(x)F^{\mu\nu}(x) - \frac{4\pi}{c} A_\mu(x) j^\mu(x).$$

Evaluating the Euler-Lagrange-equations:

$$\partial_\mu \frac{\partial \mathcal{L}}{\partial(\partial_\mu A_\nu(x))} - \frac{\partial \mathcal{L}}{\partial A_\nu(x)}=0,$$

leads to the inhomogenious Maxwell equations (Gauss's Law & Amperé's Law):

$$\partial_\mu F^{\mu\nu}(x) = \frac{4\pi}{c} j^\nu(x).$$

The homogenious Maxwell equations (divBzero & Faraday's Law) are given by an algebraic tensor relation for the field strength tensor:

$$\partial^\mu F^{\nu\rho}(x) + \partial^\nu F^{\rho\mu}(x)+ \partial^\rho F^{\mu\nu}(x) = 0.$$

Of course, you now have to insert back the definitons for $F^{\mu\nu}$ in terms of the electric and magnetic field to see the real equivalence. You can see the matrix form of it on this Wikipedia article.

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A lot was derived from experimentation, measuring magnetic fields involving electric currents and observed rates of change etc etc. Maxwell deduced, i.e. figured out that light and em were likely the same thing and proved it by measuring 2 constants (magnetic permeability and dielectric permittivity) and relating these 2 constants to calculate the speed of light, which he then verified was the same!

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