Opposing force as a conducting loop enters magnetic field

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I think it has to do with inductance. E.g. B-field points from right to left; you move conducting loop left to right.
As you move it left to right, the magnetic flux through the loop is getting stronger; in other words, the change in magnetic flux is toward the left. Faraday's Law of Inductance (and I believe one of Maxwell's equations) states that the induced electromotive force and subsequent current will act OPPOSITE to the direction of the change in magnetic flux; in other words, d(magnetic flux) = -E (E is electromotive force, or induced voltage). So for the above example, if d(magnetic flux) points to the left, a current will be induced such that it will produce a magnetic force to the right (using the thumb rule for closed loop circuits, you'll see this induced current will be counterclockwise).
 
Yup, just remember that magnetic flux is proportional to magnetic field strength and area.

And induced emf = - change in magnetic flux/ change in time.

The negative indicates that it's in the opposite direction.

So a loop going into a magnetic field will experience a change in A (it increases as it enters the field) leading to an emf, while it's in the field there's no current because there's no change in magnetic field or area, and as it's leaving the area decreases so there's a flux and it's in an opposite direction to that initial current.
 
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