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This question is because something made me confused. I always thought that the idea behind electrodynamics was to postulate some things, like Coulomb's law in electrostatics and so on, and then manipulate those things to derive Maxwell's Equations. After that we could then use Maxwell's Equations at will.

Now, I'm confused because I've seen people assuming the equations. Namely saying that the electromagnetic field is a pair of vector fields $(E,B)$ such that $E$ and $B$ satisfies Maxwell's equations. Then with this the consequences are derived. But this seems strange to me: if you start by Maxwell's Equations, is because you already know them, but to know them, you needed to start by assuming other things, instead of the equations themselves!

My thought was: historically we start really with Coulomb's law and so on, and then we find out that there's a system of $4$ differential equations that completely specify the magnetic and electric field. Then since we found this out, practically speaking, we simply forget about the rest and restart from these equations. Now, why do we do this? Why do we start from the equations rather than deriving them?

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Essentially a duplicate of physics.stackexchange.com/q/3618/2451 and links therein. –  Qmechanic Sep 15 '13 at 14:36
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Because history is a winding path filled with accidents and coincidences. Teaching things in exactly the same order as history is simply misguided. –  genneth Sep 15 '13 at 16:18
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@Qmechanic it's not a duplicate, the OP is asking why Maxwell's equations have replaced Coulombs law, even though Coulombs law was "postulated" to start with. –  Larry Harson Sep 15 '13 at 17:38
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Axiomatizations are not unique. For example, you can take Euclid's postulates and prove the Pythagorean theorem. But you can also take Euclid's first four postulates, get rid of the parallel postulate, and add in the Pythagorean "theorem" as a postulate. Then the parallel postulate becomes a theorem. –  Ben Crowell Sep 15 '13 at 20:00
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2 Answers

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In general physics course we assume Maxwell's equations as the result of many experiments. After that, in field theory we build the Lagrangian which satisfies the Maxwell equations. We also can build non-linear field theories of EM interactions, but there is a requirement to getting of the Maxwell equations in the limit of weak fields. So these methods aren't connected with the derivation.

But we can derive the equations by using some postulates, which generalize experimental facts. The number of postulates should be reduced to a minimum. Maxwell's equations can be earned from the Coulomb's law, Special relativity theory and superposition principle (more details you can see in my answer on this question).

The other question is why do we need derivation of equations instead of postulating them. They satisfy the experiments, so that's enough for using them in practical cases. The postulating them is not much worse then deriving in this situation.

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Coulomb's law was published in 1785 by Coulomb, and it's as relevant today as it was then ,because no experiment as contradicted it.

Additional laws were added over the years including Ampere's law in 1826, culminating in Maxwell's equations in 1861-1862, but given their modern form by Heaviside later on.

So no, Maxwell's equations haven't replaced Coulomb's law, they've included it in Gauss's law which is equivalent to Coulomb's law: $$\nabla\cdot\vec E= \rho/\epsilon_0$$

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