Final answer:
To produce zero magnetic field at the center of the loop, the current in the loop must flow in the direction opposite to the one determined by the right hand rule for the wire. The magnitude needed for this current can be calculated by equating the loop's magnetic field magnitude to that of the wire's and solving for the current.
Step-by-step explanation:
To produce zero magnetic field at the center of a single current-carrying circular loop of radius R placed next to a long, straight wire, we need to find the direction and magnitude of the current in the loop that counteracts the magnetic field from the wire at the loop's center.
Using the right hand rule 2 (RHR-2), the magnetic field direction created by the long wire can be determined. With the thumb pointing in the direction of the current I in the wire, the fingers will curl in the direction of the magnetic field, which in this case would be counterclockwise when viewed from the other side of the loop facing the wire. To negate this field at the center of the loop, the loop's current must produce a magnetic field in the opposite direction. Therefore, for the loop's magnetic field to be clockwise, the current must be flowing in the direction opposite to what the right-hand rule would suggest for the external wire.
The magnitude of the magnetic field at the center of the circular loop is given by B = (MOI) / (2R) where MO is the permeability of free space, I is the current through the loop, and R is the radius of the loop. To make the total magnetic field zero at the center, this field must be equal in magnitude but opposite in direction to the wire's magnetic field there. By setting these two magnetic fields equal and solving for the loop current I, we establish the magnitude needed to cancel out the wire's field at the loop's center.