A wire \(1.0 \mathrm{mm}\) in diameter carries \(5.0 \mathrm{A}\) distributed uniformly over its cross section. Find the field strength (a) \(0.10 \mathrm{mm}\) from its axis and (b) at the wire's surface.

When we're dealing with magnetic fields and electrical currents, one pivotal principle comes into play: Ampere's Law. This law is a cornerstone in the study of electromagnetism and provides a way to calculate the magnetic field created by electric current.

Ampere's Law states that the line integral of the magnetic field (B) around a closed loop is equal to the permeability of free space (μ_0) multiplied by the electric current (I) passing through the loop. Mathematically, it’s expressed as ∮ B · dl = μ_0I, where the integration path is around the loop and 'dl' is an infinitesimal element of the loop path.

In the context of a wire carrying a uniform current, Ampere's Law simplifies to a more straightforward relationship. If we're looking at a point inside the wire and at a distance 'd' from the center, Ampere's Law can be adapted to calculate the magnetic field created by the current at that point.

The magnetic field strength, represented by the symbol B, is a measure of the magnitude of the magnetic field at a point in space. It's a vector quantity that points in the direction a north pole of a magnetic compass would align to.

Within a wire carrying a steady current, this magnetic field strength at a distance 'd' from its axis can be found using a form of Ampere's Law. The strength of the magnetic field inside the wire varies linearly with the distance from the center due to the uniform distribution of current. So, the closer you are to the center of the wire, the weaker the magnetic field you'll encounter. Conversely, at the surface of the wire, the magnetic field reaches its maximum strength relative to points inside the wire.

Using the formula derived from Ampere's Law, B = μ_0Id/(2πr^2), for points inside the wire and B = μ_0I/(2πr) for the surface, one can calculate the precise strength of the magnetic field at specific locations relative to the wire's radius.

An essential constant in electromagnetism is the permeability of free space (μ_0), also known as the magnetic constant. This scalar value represents the measure of resistance that is encountered when forming a magnetic field in a classical vacuum.

μ_0 has a standard value of approximately 4π × 10^{-7} T·m/A (Tesla meters per ampere), a figure that appears frequently when you're dealing with Ampere's Law and magnetic fields. It lays the ground for magnetic field calculations and is one of the defining aspects of how electric current creates a magnetic field.

Understanding the role of the permeability of free space is vital, especially when working with Ampere's Law or when trying to calculate the magnetic field strength inside materials with different magnetic properties. It fundamentally ties together electric current and the resulting magnetic field in free space or other media.