A conventional current of 6 A runs clockwise in a circular...

A conventional current of 6 A runs clockwise in a circular loop of wire in the $x y$ plane, with center at the origin and with radius $0.08 \mathrm{m}$. Another circular loop of wire lies in the same plane, with its center at the origin and with radius $0.02 \mathrm{m} .$ How much conventional current must run counterclockwise in this smaller loop in order for the magnetic field at the origin to be zero?

Key Concepts

Magnetic Field of a Circular Current Loop

A circular current loop produces a magnetic field at its center that is directly proportional to the current flowing through it and inversely proportional to the radius of the loop. The standard formula, derived from the Biot–Savart law, is B = ??I/(2R), where ?? is the permeability of free space, I is the current, and R is the radius of the loop. This relationship is fundamental in analyzing magnetic fields generated by current distributions with circular symmetry.

Right-Hand Rule

The right-hand rule is a convention used to determine the direction of the magnetic field produced by a current flow in a loop. By curling the fingers of the right hand in the direction of the conventional current, the extended thumb points in the direction of the magnetic field along the axis of the loop. This rule is essential for establishing the vector directions of magnetic fields in electromagnetic problems and for understanding how opposing current directions can create fields that cancel each other.

Superposition Principle

The principle of superposition states that the net magnetic field at any point is the vector sum of the magnetic fields produced by each individual current-carrying element. In contexts where multiple loops or current paths are present, this principle allows one to analyze the contribution of each loop separately and then combine them to determine the overall magnetic field. It is especially useful in problems where fields from loops with currents in opposite directions counteract each other to achieve cancellation.

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