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I'd like to know what are the basic equations of electromagnetism, that can be used to formulate all the other laws and equations.

Those basic equations I can think of are Maxwell equations, Lorentz force equation and Coulomb's law. What are the other fundamental laws and equation?

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You just need Maxwell's equations and the Lorentz force law. Coulombs law can be derived from Maxwell's equations.

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See for details behind that statement – Floris Jul 4 '14 at 0:33
In addition, Coulomb's Law doesn't apply for moving charges. – Shivam Sarodia Jul 4 '14 at 2:34
@Draksis There's no loss of generality there, as you can always choose a frame in which the charges are stationary. Coulomb's law still holds as long as you Lorentz boost. – Ryan Lafferty Jul 4 '14 at 19:49
@RyanL Would it always be possible to do so, however? For example, in a system with charges moving with different velocities, there would be no single frame for which all charges are stationary. – Shivam Sarodia Jul 4 '14 at 20:04
If you do a Lorentz boost on Coulomb's law, you get Biot-Savart's law which you would be using anyway for moving charges. So trivially, yes. Don't do that, it's nasty but my point is Coulomb's law always holds in some chosen frame. It just isn't covariant. – Ryan Lafferty Jul 4 '14 at 20:23

The basics about the direction of force and field comes from the "Fleming's Left Hand Rule" and the "Maxwell's Corkscrew Rule". In addition to these the Lorentz force law, i.e. F=q[E+(vxB)] gives the force on a charge moving through a magnetic and electric field [Neglect E if electric field is absent.] The Biot- Savart Law gives the relation between current and magnetic field.

The Ampere's Circuital Law also plays a pivotal role in finding out magnetic field due to current through an enclosed area.

Coming to electric field due to magnetism[commonly called, Electromagnetic Induction], there exists the Lenz's Law. The concepts of motional emf and Eddy currents are essential for understanding the topic.

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