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Are there other equations that we can predict Faraday's law from? I know that each of Maxwell's equations are 'fundamental', but I feel like Gauss's law and Ampere's Law are very "nice", and for some reason or another I don't feel the same about Faraday's law. I know that, obviously, any other way of 'deriving' Faraday's law will in effect just be another way of restating Faraday's law, but is there any other way besides noting the fact that changing electric fields and changing magnetic fields are related? If this is the only way, can someone provide a quick run through of the process that went into developing this law?

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The Faraday law in terms of fields is mathematical formulation of law of electromagnetic induction, which was discovered by Faraday in his experiments. He found that changing magnetic flux through an electric circuit (magnetic field being due to magnet or due to other current carrying circuit) is accompanied by additional electromotive force in the circuit. Maxwell formulated this mathematically in a form close to ours (using the vector calculus).

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