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Ampère's law is a fundamental principle in electromagnetism. It's a practical way to determine the relationship between electric currents and magnetic fields. According to this law, the magnetic field around a closed loop is proportional to the electric current flowing through the loop. This relationship is expressed mathematically as:
\[ \oint \mathbf{B} \cdot d\mathbf{l} = \mu_0 I \]
Where:

• \( \oint \mathbf{B} \cdot d\mathbf{l} \) represents the line integral of the magnetic field \( \mathbf{B} \) around a closed path.
• \( \mu_0 \) is the permeability of free space.
• \( I \) is the current enclosed by the path.

Ampère's law allows us to calculate the magnetic field generated by a current-carrying wire, as seen in the given problem. It shows that the magnetic field's strength decreases with distance from the wire, emphasizing the inverse relationship between distance and field strength.

A current-carrying wire is a common source of magnetic fields in physics. When an electric current flows through a wire, it generates a magnetic field that encircles the wire. The direction of this magnetic field is determined by the right-hand rule:

• Point your thumb in the direction of the current.
• Your fingers will curl in the direction of the magnetic field lines.

The magnetic field is strongest close to the wire and weakens as you move away. This concept is essential in explaining why at a distance 3R from a wire, the field is reduced to one-third of its strength at distance R. Understanding this interaction helps in designing electrical devices and managing electromagnetic fields in technology.

The permeability of free space, denoted as \( \mu_0 \), is a physical constant essential in electromagnetism. It quantifies the ability of a vacuum to support magnetic field lines. Its value is approximately \( 4\pi \times 10^{-7} \text{ T} ext{m/A} \) (Tesla meter per Ampere), which is fundamental in the equations that describe magnetic fields:
\[ B = \frac{\mu_0 I}{2 \pi r} \]
Here, \( \mu_0 \) helps determine how effective a magnetic field is generated by a current in free space. It is a cornerstone in calculating the magnetic field effects around current-carrying conductors. Understanding \( \mu_0 \) is crucial for accurately solving problems related to magnetic interactions, like the one given, where it helps link the magnetic field to the current and the geometry of the system.

The magnetic field formula for a long, straight, current-carrying wire is given by:
\[ B = \frac{\mu_0 I}{2 \pi r} \]
This formula helps calculate the magnetic field \( B \) at a distance \( r \) from the wire. Several key components determine the field's strength:

• \( \mu_0 \): Permeability of free space, which reflects how easily a magnetic field can pass through the vacuum.
• \( I \): The current flowing through the wire, which is directly proportional to the strength of the magnetic field.
• \( r \): The distance from the wire, which is inversely proportional to the magnetic field strength.

Knowing how to manipulate this formula is crucial for understanding various practical applications, from designing circuit boards to magnetic resonance imaging (MRI) in medical fields. It illustrates the foundational relationship between current, distance, and magnetic field strength.