A 2.2 -m-long wire carrying 3.5 A is wound into a tight coil \(5.0 \mathrm{cm}\) in diameter. Find the magnetic field at its center.

When we delve into the fascinating world of magnetism, Ampere's Law is a beacon that guides our understanding. It's a fundamental concept in electrodynamics, elegantly linking magnetism to electric current. Ampere's Law states that for any closed loop path, the sum of the length elements times the magnetic field in the direction of the length element is equal to the permeability of free space times the electric current enclosed by the loop.

To put it simply, Ampere's Law helps us calculate the magnetic field produced by an electric current. It provides a way to find the magnetic field in a region when you know the current distribution. With this key principle in mind, we can begin to see that the shape of the wire, the current flowing through it, and the space or medium through which it travels all influence the magnetic field that's generated.

Calculating the magnetic field, especially in the case of a wire coil, is essential for many applications in electromagnetism. The calculation combines physical constants with the properties of the coil itself. The constant \(\mu_0\), known as the permeability of free space, is a measure of how much resistance is encountered when forming a magnetic field in a classical vacuum.

To relate this to our exercise, knowing the current (I) and the geometry of the coil (its radius, in particular), we can use a specific form of Ampere's Law to compute the magnetic field at the coil's center. The formula \(B = (\mu_0 \times I) / (2 \times r)\) simplifies the process, focusing on the proportional relationship between the magnetic field (B) and the current (I), and an inverse relationship with the radius (r) of the coil.

The magnetic field at the center of a wire coil is a unique point where the field's strength can be considered quite uniform. This particular magnetic field arises from the symmetry of a current-carrying loop. Each segment of current provides a small magnetic field contribution, and due to the shape of the coil, all these contributions add up to a stronger, centralized field at the center of the coil.

In the exercise at hand, the wire length, current, and coil diameter are given to us. To apply the relevant formula, we first need to change the diameter to radius and the centimeters to meters for consistency in the units. The tightness of the coil implies that we can regard it as a series of loops closely wound together, increasing the total magnetic field at the center due to the combined effect of each turn of the wire with current flowing through it.