This law is named after German scientist H. F. E. Lenz.

It states:

The direction of any magnetic induction effect is such as to oppose the cause of the effect.

Lenz's Law prevents funny things from happening, such as using magnets to accelerate objects to energy levels beyond that of the original energy in the system. In other words, it is just another aspect of the conservation of energy.
Despite its apparent ambiguity, Lenz's Law can be used to predict the way that a current will be induced in a conductor. So, keeping in mind that "the direction of an induced e.m.f. is such that it will always try to oppose the change in magnetic flux producing it", let's consider an example:

Let's say you have a bar magnet on the left, with its N pole pointing towards a solenoid on the right, and moving towards the solenoid. In this case, the change in magnetic flux is caused by the relative motion of the magnet and the solenoid, so the solenoid is "cutting through" the lines of magnetic flux from the magnet (since these are not all parallel to the direction of motion). Hence, an e.m.f. will be induced. We can use Lenz's Law to determine in what direction the current will flow in the solenoid.

In order to oppose the change in magnetic flux, i.e. reduce the rate at which the solenoid is "cutting through" field lines, the relative velocity of the solenoid with respect to the magnet has to decrease. So, the solenoid will have an e.m.f. induced in it that causes the magnet to slow down (relative to the solenoid; the solenoid could also speed up in the direction of motion of the magnet, it makes no difference). In other words, the solenoid repels the magnet. Therefore the current induced in the solenoid will effectively make the end of the solenoid nearer to the magnet a N pole. In other words, the magnetic field lines will go out of the solenoid at that end.

Almost there. We know that the left end of the solenoid must have magnetic field lines coming out of it, so we can figure out the direction of the current flowing in the solenoid using the right-hand grip rule: you curl your right hand into a fist, and stick your thumb out. Your thumb shows the direction of the magnetic field lines inside the solenoid, and your curled fingers show the direction of flow of current in the solenoid, the tips of your fingers being the ends of the "arrows" showing the direction of the current. (This exact hand position also works for a current in a straight wire: in that case, the thumb shows the direction of the current, and the curled fingers show the direction of the circular magnetic field lines.) In our example, the field lines inside the solenoid go from right to left, so point your thumb to the left. Finally, looking at your fingers, you now know the direction in which the current flows in the solenoid. Incidentally, the direction of current indicated by the right hand grip rule is conventional current, so the e.m.f. will actually be in the opposite direction.

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