Final answer:
The force between two closed current loops can be derived using Biot-Savart and Lorentz force laws, with Newton's third law ensuring mutual forces are equal and opposite. Calculating this for specific arrangements of loops requires complex calculations.
Step-by-step explanation:
The general expression for the force exerted on one current loop by another can be derived from the Biot-Savart Law and the Lorentz force law. If we have two closed loops, labelled 1 and 2, carrying currents I1 and I2, the force on loop 1 by loop 2 depends on the magnetic field produced by I2 and the orientation and position of loop 1 relative to that field. The law of action and reaction (Newton's third law) ensures that the force on loop 2 due to loop 1 is equal and opposite to the force on loop 1 due to loop 2.
For two specific circular loops with radius R, where one loop is in the XY plane and the other is in the YZ plane intersecting at the origin, calculating the force between them would involve complex integration as their magnetic fields would vary with both distance and position. The exact analytical expression for this force is not straightforward and typically demands advanced electromagnetic theory for precise computation. However, it's important to note that based on symmetry and the right-hand rule, the net force for symmetrical arrangements will comply with Newton's third law, signifying the forces are equal in magnitude and opposite in direction.