Introduction:
When a square loop carrying a current i is placed near a long straight conductor carrying a current I, the loop experiences a net force due to the magnetic field produced by the current in the conductor. This force can be determined using Ampere's law and the right-hand rule.

Explanation:

Magnetic Field:
The long straight conductor XY carrying a current I produces a magnetic field around it. According to Ampere's law, the magnitude of the magnetic field at a distance r from the conductor is given by:

B = (μ₀ * I) / (2π * r)

Where μ₀ is the permeability of free space.

Force on a Current-Carrying Wire:
When a current-carrying wire is placed in a magnetic field, it experiences a force. The force can be determined using the right-hand rule. If the thumb of the right hand points in the direction of the current and the fingers wrap around the wire, the direction of the force is given by the direction in which the palm faces.

Net Force on the Square Loop:
To determine the net force on the square loop ABCD, we need to consider the individual forces acting on each side of the loop.

1. Force on side AB:
The current i in side AB of the loop produces a magnetic field. This magnetic field interacts with the magnetic field produced by the current I in the conductor XY. The force acting on side AB of the loop can be determined using the right-hand rule. The direction of the force depends on the relative orientations of the magnetic fields. If the magnetic fields are parallel, the force is attractive, and if they are anti-parallel, the force is repulsive.

2. Forces on sides BC, CD, and DA:
Similar to side AB, each side of the loop experiences a force due to the interaction between the magnetic fields. The direction of the forces on these sides depends on the orientations of the magnetic fields.

3. Net Force:
The net force on the square loop is the vector sum of the forces acting on each side. If the forces on opposite sides are equal and in the same direction, they cancel each other out, resulting in a net force of zero. However, if the forces on opposite sides are not equal or in the same direction, there will be a net force on the loop.

Conclusion:
In summary, when a square loop carrying a current i is placed near a long straight conductor carrying a current I, the net force on the loop depends on the orientations of the magnetic fields produced by the current in the loop and the conductor. The individual forces on each side of the loop can be determined using the right-hand rule, and the net force is the vector sum of these forces.