Where is this circular loop attracted? [closed]

Question

A circular loop carrying current "I" in anti-clockwise direction is placed between two straight and parallel wires each carrying current I. Ignoring the force of the straight wires on each other, where is the circular loop attracted towards?

I've been struggling with this question and need some conceptual help. First, I thought of using Fleming's left hand rule.

On the left extreme of the loop, if we use the same with the inward magnetic field of the straight wire on the left, we find that a force acts on the circular loop towards the right side. Doing the same on the right hand extreme of the loop with the outward magnetic field of the right hand straight wire again gives a force towards the right hand side acting on the circular loop. So I think that the loop is attracted to the right side wire.

But I'm not sure if my reasoning is correct since I consider the force at the right and left extremes of the loop only and that too with the magnetic field of only the nearer wire and not the faraway one.

Hence I will appreciate some help with the problem.

Answer 1

Your conclusion is correct.

To account for the fields of the parts of the wire that are not at the extremes, first convince yourself that the net magnetic field from the wires points out of the screen in the right half of the region between the wires; and points into the screen in the left half.

Once you have convinced yourself of this, consider the force on a small bit of the wire that is closer to the right-hand wire than the left-hand wire, such as the green one in the above diagram. Using the Lorentz force law (and the right-hand rule that is associated with it), you can hopefully see that the force on this piece of wire points up and to the right.

Similarly, for pieces of wire on the left half of the loop (such as the red or blue ones in the above diagram), the force on that piece of the wire will also experience a force whose horizontal component also points to the right. In fact, all of the elements of the loop will experience a force whose horizontal component points to the right; and so the net force on the loop must point to the right as well. (An element right on the center line would experience zero force, but this doesn't change our overall conclusion.)

Note also that by looking at the individual elements of the wire this way, we can also conclude that the net vertical force on the loop is zero. For example, the upwards component of the force from the green piece of the wire is exactly cancelled by the force on its "mirror image" (the blue piece). In fact, we can imagine pairing all of the forces off in this way, and from this we can see that all of the vertical forces cancel out.