# Intuition of Maxwell's Equations [duplicate]

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Is there an intuitive explanation for Maxwell's equations?
I know they are axioms but is there a logical understanding of why instead of mathematical. Both forms don't explicate the scientific reasoning behind them to me.

I would appreciate a non- or minimally mathematical approach to them.

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• One, for example, is also known as Gauss' Law for magnetism and states the nonexistence of magnetic monopoles. If you were to find one you'd disprove that law, however there are some valid arguments as to why it should not exist. I suggest you to look into Faraday's law, Ampère's law and Gauss' Law to understand what's the physical meaning of the equations. – Bonsay Jul 17 at 14:11
• Possible duplicate of Physical meaning of Maxwell's equations and origin of EM waves – Aaron Stevens Jul 17 at 14:12
• They represent conservation of charge under the Lorentz transformation as a consequence of the U(1) symmetry (per the Noether theorem). Charges moving in space create a magnetic field; charges moving in time create an electric field. en.wikipedia.org/wiki/Electromagnetic_tensor#Relativity – safesphere Jul 17 at 14:54
• I know they are axioms Not really. Physical theories aren't axiomatic mathematical systems, and even in a mathematical theory, there is no well-defined way to decide which things are axioms and which are theorems. – Ben Crowell Jul 17 at 15:57
• @Bonsay There are also good reasons why magnetic monopoles "should" exist. The biggest one is Dirac's argument that the existence of magnetic monopoles leads to charge quantization, which is a phenomenon that otherwise doesn't have a good explanation. (But that's going pretty far afield from the OP's question.) – sasquires Jul 17 at 16:13

## 2 Answers

The two equations involving the divergence aren't dynamical (they have no time derivatives), and if they're satisfied initially, they're automatically satisfied at all later times. They tell us about the sources and sinks of the fields.

The two equations involving the curl have time-derivative terms and a current term.

The time-derivative terms describe electromagnetic induction. Their signs are opposite, which is what allows negative feedback so we can have oscillating electromagnetic waves.

The current term describes how moving charges create magnetic fields. It has to be there because of Lorentz invariance, i.e., if we had known about electric fields but not about magnetic fields, relativity would have forced us to invent magnetism.

They’re not axioms: They’re experimental results.

Coulomb, Faraday, etc did a lot of experimental work to observe and systematize the underlying phenomena. Maxwell then reformulated them (though not in the modern form) and added the displacement current term which itself was later confirmed experimentally.

So the “why” historically comes down to “because people observed this”

Today, the underlying “why” is “because these emerge from the quantum field theory of QED” and even fro electroweak interactions. But that’s a very long leap to make intuitive.

So if you want an intuitive understanding of Maxwell’s equations themselves, the best place to look is at the experiments that underlies them. It’s a lot to put in an Answer, because it’s a lot of physics.

For example, forces on small charges led to Coulomb’s law, which led to the idea of an electric potential hence electric field and Gauss’ law.

• Well ... they are axioms as far as E&M calculations are concerned. – garyp Jul 17 at 16:41