Final answer:
The magnetic field inside a 2.8 mm-diameter copper wire carrying a 31 A current, 0.50 mm below the surface, can be calculated using Ampère's law, with the formula B = (μ_0 ∙ I ∙ r) / (2 ∙ π ∙ R^2). By substituting the given values and doing the calculation, one can find the magnetic field at the specified location.
Step-by-step explanation:
To determine the magnetic field inside the 2.8 mm-diameter copper wire carrying a 31 A current, we use Ampère's law, which relates magnetic fields to electric currents. Because the current is uniformly distributed, the magnetic field strength at a point within the wire depends only on the current enclosed by the path that is at the radial distance from the center of the wire. The formula derived from Ampère's law for the magnetic field inside a long, straight, and uniformly carrying current wire is B = (μ_0 ∙ I ∙ r) / (2 ∙ π ∙ R^2), where μ_0 is the permeability of free space (4π × 10^{-7} T∙m/A), I is the current, r is the radial distance from the center of the wire, and R is the wire's radius.
Given that the radial distance from the center is 0.50 mm away from the surface (thus 1.4 mm - 0.5 mm = 0.9 mm from the center), the radius of the wire R is 1.4 mm (2.8 mm/2), and the current I is 31 A, we can substitute these values into the formula to find the magnetic field inside the wire at the specified distance from the surface. Remember to convert all millimeter measurements to meters in the calculation.