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When a magnetic north(N) pole is placed close to an coil with out energy source, the coil behaves like an magnet and we find a North pole which opposes the magnet bar which was approximating onto the coil.

My question is, what conditions and mathematcal treatment assure us about the creation of the pole on the coil.

Please write if you have any confusion to get the question.

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I think you're talking about the induced current in the wire: a changing magnetic field (caused by the approaching magnet, which is by the way NOT just a magnetic north pole: those don't exist) causes a current to run through the coil. This is known as Faraday's Law, I believe. Now, a current will always produce a magnetic field, and the induced current is such that it opposes the change in magnetic flux (Lenz' Law). In the case of an approaching magnet, with the north pole closer to the coil than the south pole, the induced field will 'fight' the increase in flux.

Mathematically, we have Faraday's law $$\nabla \times E=-\frac{dB}{dt}$$ this is essentially the math that shows that there will be a current due to the changing magnetic field. Furthermore, this current gives rise to a magnetic field: $$\nabla \times B=\mu_0J+\frac{dE}{c^2dt}$$

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