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Ampere's Law is a fundamental principle used to relate the magnetic field around a closed loop to the current passing through it. Essentially, Ampere’s Law states that the magnetic field integrated along a closed path is equal to the permeability of free space times the current passing through the path. This law is mathematically expressed as:

• \( \oint \mathbf{B} \cdot d\mathbf{l} = \mu_0 I \)

Here, \( \mathbf{B} \) is the magnetic field, \( d\mathbf{l} \) is an infinitesimal element of the path, \( \mu_0 \) is the permeability of free space, and \( I \) is the current enclosed by the loop.
In situations involving long, straight conductors, Ampere's Law simplifies the calculation of magnetic fields, especially for symmetrical configurations like cylindrical wires. By choosing an Amperian loop that fits the symmetry of the wire, the magnetic field can be calculated accurately.

The current flowing through a conductor, such as a wire, is the rate at which electric charge passes through a cross-sectional area of the wire. It is usually denoted by \( I \) and measured in amperes (A). In this problem, we're dealing with a straight tungsten wire carrying a current of \( 3.5 \) A.
This current generates a magnetic field around the wire due to the movement of charges. Current in a wire can be visualized as a stream of electrons moving through the conductor, and the amount of current depends on two main factors:

• The number of charges passing through the wire per unit time.
• The charge each of these particles carries.

Understanding the relationship between current and magnetism is key to calculating the magnetic field within or around the wire.

Permeability of free space, symbolized as \( \mu_0 \), is a constant that quantifies the ability of a vacuum to sustain a magnetic field. The value of \( \mu_0 \) is approximately \( 4\pi \times 10^{-7} \) T·m/A.
This constant is vital in calculating magnetic fields using Ampere's Law. It reflects how much magnetic field is produced in a vacuum by a given current, serving as a bridge between electric currents and the magnetic fields they generate.
In essence, permeability of free space helps understand how electromagnetic fields propagate in a vacuum, playing a crucial role in many electromagnetic calculations.

To calculate the magnetic field inside a wire, we use the formula derived from Ampere's Law:

• \( B = \frac{\mu_{0} I r}{2 \pi R^2} \)

Here, \( B \) is the magnetic field, \( I \) is the current, \( r \) is the distance from the central axis, and \( R \) is the wire's radius stored as \( 1.2 \times 10^{-3} \) m for this particular problem.
This formula enables the understanding of how the magnetic field varies within the wire. The magnetic field is directly proportional to the current in the wire and the radial distance \( r \), yet inversely proportional to the square of the radius \( R \).
By substituting known values into this equation, you can find the magnetic field's strength at any point inside the wire, as demonstrated in the given problem. This step-by-step approach ensures clarity in calculation.