Why ONLY Maxwell's equations are the basic equations of electromagnetism?

Question

In electromagnetism we say that all the electromagnetic interactions are governed by the 4 golden rules of Maxwell. But I want to know: is this(to assume that there is no requirement of any other rule)only an assumption, a practical observation, or is there a deeper theoretical point behind it? Could there be a deeper theory behind assuming that there is not requirement of rules other than Maxwell's equations?

Answer 1

In electromagnetism we say that all the electromagnetic interactions are governed by the 4 golden rules of Maxwell. But I want to know that is this only an assumption

It is not an assumption, it is an elegant way of joining the diverse laws of electrictity and magnetism into one mathematical framework.

or a practical observation

The laws of electricity and magnetism were described mathematically by fitting observations and always being validated, i.e. correct, in their predictions. Maxwell's equations not only incorporated them but also by unifying electricity and magnetism mathematically give predictions that have never been falsified.

So yes, they are a mathematical model fitting observations, a very elegant model.

or there exist any theoretical point behind it?

Physics is about observations and the derivation of mathematical models, theories, that will fit them and will also predict new observations to be measured and evaluate the theory. Physics is not about philosophy or mathematics, it is about describing nature using mathematics as a tool.

If there exists a "theoretical point" it is that theoretical physicists try to unify in one mathematical model all the known observations, i.e. continue on what Maxwell has done in unifying electricity and magnetism, by unifying the weak with the electromagnetic, and proposing a unification with the strong in similar mathematical frameworks. The goal being in unifying also gravity, all four forces in one mathematical model

Answer 2

The Maxwell equations only approximately describe electromagnetism, even in a pure vacuum. This is a consequence of quantum electrodynamics. One can derive corrections to the Maxwell equations; this was first done by Heisenberg an Euler in the regime where the fields only change appreciably over distances much larger than the electron Compton wavelength, see here.

Answer 3

Depending on how "basic" you consider an equation to be to electromagnetism, you could consider other equations to be important enough to be thought of as basic, given the type of situation.

For instance, when dealing with electromagnetism in media (typically linear media), the Constitutive Relations also apply and are necessary:

$$\overrightarrow{D} = \varepsilon_{0} \overrightarrow{E} + \overrightarrow{P}$$

$$\overrightarrow{H} = \frac{\overrightarrow{B}}{\mu_{0}} - \overrightarrow{M}$$

Or, if you happen to be relating currents to charges, you may want to use the Continuity Equation (although you can derive this by taking the divergence of Ampere's Law with the Maxwell Correction):

$$\nabla \cdot \overrightarrow{J} + \frac{\partial \rho}{\partial t} = 0$$

Furthermore, if you are dealing with point charge with mass in addition to electromagnetic fields, the Lorentz Force Equation will be needed (although you can derive this from Newton's 2nd Law, Lagrangian Mechanics, or Hamiltonian Mechanics):

$$\overrightarrow{F} = q(\overrightarrow{E}+ \overrightarrow{v}\times \overrightarrow{B})$$

But if you are being strict, and want to have the most bare-bones version of Maxwell's Equations (in a vacuum), you can get away with only two equations, those of the vector and scalar potentials:

$$\overrightarrow{B} = \nabla \times \overrightarrow{A}$$

$$\overrightarrow{E} =- \nabla \Phi - \frac{ \partial \overrightarrow{A}}{ \partial t}$$

And by taking the divergence and curl of each of these equations, you can recover the four Maxwell's Equations.

The number of equations you need really boils down to what type of problem you are trying to solve.

Answer 4

I appreciate @ACuriousMind's comments on my question and thank him/her for pointing out the link that he/she has pointed out. I also apologize for being somewhat reluctant about mathematics in my comments when I posted the question.

The question was whether Maxwell's equations are the only equations that govern the electromagnetic interactions. With the assumption that the Lorentz force law is given (that is to say that it has been determined experimentally beyond any doubt), the question reduces to asking "Whether Maxwell's equations determine the electromagnetic fields completely?". Now, I think the answer is obvious and is answered by the famous and beautiful Helmholtz theorem. With all due respect, I wonder why no other answers chose to mention this simple and conclusive response and instead some chose to patronize the OP about how science works and what physics is about.

The Helmholtz theorem states that if $\nabla \cdot \vec{M}= U$ and $\nabla \times \vec{M}=\vec{V}$, $U$ and $\vec{V}$ both go to zero as $r \to \infty$ faster than $\displaystyle\frac{1}{r^2}$, and $\vec{M}$ itself goes to zero as $r \to \infty$ then $\vec{M}$ is uniquely and consistently determined in terms of $U$ and $\vec{V}$.

Once identifying $\vec{M}$ with $\vec{E}$ and then with $\vec{B}$, it is clear that the Helmholtz theorem dictates that the electromagnetic field is uniquely determined by Maxwell's equations and thus, there cannot be any additional new law of electrodynamics. Because there is nothing left to be described.