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So according to Faraday's law, when there is a change in magnetic flux, an EMF is induced. The reason why an EMF is induced is because there is an electric field produced. So my question is, how exactly does a change in magnetic fields causes curling electric fields to be produced?

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Faraday's law , is called a "law" because it fits laboratory observations and allows for the mathematical modeling that fits the data, and is predictive of new.

Before the brilliant unification of electricity and magnetism in one mathematical form , by Maxwell, observations of electricity and magnetism were gathered into "laws" each leading to a mathematical model.

With Maxwell a general set of differential equations was established , unifying the observations of electricity and magnetism, using these laws as "observational axioms", to pick up those solutions of the differential equations that fit observations and data.

So mathematically the "laws" are tied up in the equations, and the success of Maxwell's theory in describing and predicting data is great.

how exactly does a change in magnetic fields causes curling electric fields to be produced?

Within the consistent solutions of the equations, one can say that it is due to the mathematics. The fact remains that the equations developed in order to fit data and be predictive of new ones by encompassing the existence of these laws.

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