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In the case of Ampere's law, when the electric field at the area of the loop is changing, we add an extra term of displacement current. Likewise, do we also have to add an extra term in the Biot-Savart law equation to obtain the magnetic field at a point or is the law complete in itself?

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Essentially, Ampere's Law has was altered to account for displacement current in order to make it universally true. Biot-Savart was not. As far as practical physics, Biot-Savart is only a viable method to calculate magnetic fields when you are only dealing with current-carrying wires and nothing of greater consequence.

That is not to say that it's impossible to alter Biot-Savart to account for other phenomena, but in that case, you would likely choose to simply use Ampere's Law instead.

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  • $\begingroup$ in that case, you would likely choose to simply use Ampere's Law instead. This doesn't really make sense. Ampere's law doesn't give a direct method for finding a field. It's just a differential equation that the field has to satisfy. Finding a solution is a different matter. $\endgroup$ – user4552 Dec 31 '19 at 21:03

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