There are three ways of inducing current in a loop/coil of wire as shown in my book. We can have a magnet approach a coil of wire, or a wire approaching a magnet. Both can be understood in the same way.
On the other hand, we can also change the magnetic flux by pushing a loop of wire into a magnetic field.
This confuses me. Two vertical lines are cutting across the field. But since they are connected, the induced current, I conjecture, would cancel each other.
I saw in a YouTube video that to determine the direction in such situations as 2, one curl the fingers of their right hand along the wire, with the thumb pointing in the direction of the field. So the curled fingers are in the direction of the current. Basically the directions of current indicated by the thumb is always opposite the direction of the changing field.
Best Answer
The rule is called Lenz's Law. You already appear to know how to determine the direction of the magnetic field due to a current in a loop, which is part of the answer. What Lenz's Law tells us is that the direction of the induced current in the loop is such that it "opposes the change in the flux".
Here's a picture I grabbed from https://web2.ph.utexas.edu/~coker2/index.files/induction.htm to illustrate this. The B-field is pointing up and increasing. So we say there is an "induced B field" opposing this increase (so it points down). The current in the loop is such that it would create this induced field according to the usual right-hand rule for B-fields due to loops.
So imagine that the B-field was decreasing instead. So if B is up the change in the B-field is down. Thus the induced B-field would point up and the current would be opposite to what is in the diagram. Here is another diagram from the same website showing some other cases. Practice working through the reasoning of Lenz's Law to verify that you see why the induced current is the in the direction indicated for each case:
A word of warning about this concept. The change in the B-field does not really induce an "induced B-field". What it really does is create a circular E-field, and it is this which drives the current in the loop. The "induced B-field" is a fictional construct which makes it easier to figure out the direction of the current.