Final answer:
The location inside the wire where the magnetic field equals a third of its maximum value is at r/R = 1/3. There is no point outside the wire where the magnetic field reaches a third of its maximum value because it monotonically decreases.
Step-by-step explanation:
To determine at what location inside the wire the magnetic field produced by the current is equal to a third of its largest value, we need to apply Ampère's Law. Inside a conductor carrying uniform current, the magnetic field B increases linearly with the distance r from the center of the wire due to the proportion of current enclosed. Thus, the magnetic field is given by B = μ_0 J r / 2, where J is the current density and μ_0 is the permeability of free space. Since the value at the surface (r=R) will be maximum (B_max), for B to be a third of B_max, we must have (1/3)B_max = (μ_0 J r / 2). Solving for r, we find r/R = 1/3.
For locations outside the wire, Biot-Savart Law or Ampère's Law show that the magnetic field decreases with 1/r. However, since the field is maximum at the surface, and we do not have an equation that varies outside the wire, we cannot directly calculate when the field will be a third without additional information on how the field varies with r beyond the wire's surface. Normally, the largest value of B is at the surface, and it decreases monotonically outside, never increasing again to allow for a location at which it is a third of its largest value.
The magnetic field for points inside the wire is proportional to the distance from the center, and for points outside the wire, it is inversely proportional to the distance from the center. However, the magnetic field does not reach a third of its largest value at any given point outside the cylindrical wire since it monotonically decreases.