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Understanding the Biot-Savart Law is fundamental for calculating the magnetic field produced by a steady current. This law relates the magnetic field \textbf{B} to the magnitude, direction, length, and proximity of the electric current. Essentially, it states that for a small segment of current-carrying wire, the magnetic field at a point in space is directly proportional to the current, inversely proportional to the square of the distance from the wire, and depends on the angle between the wire and the point in question.

The mathematical expression for Biot-Savart Law is \( dB = \frac{{\mu_0}}{{4\pi}} \frac{{Id\vec{s} \times \hat{r}}}{{r^2}} \) where \( Id\vec{s} \) is the vector representing the infinitesimal current element in both magnitude and direction, \( \hat{r} \) is the unit vector from the current element to the point of interest, \( r \) is the distance from the current element to the point, and \( \mu_0 \) is the permeability of free space.

This law provides the basis for the derivation of the magnetic field for various geometries, including a circular coil, as students must consider in this exercise.

The permeability of free space, denoted \( \mu_0 \), is a fundamental physical constant which characterizes the ability of a vacuum to support the formation of a magnetic field. Specifically, \( \mu_0 \) is the measure of the amount of resistance encountered when forming a magnetic field in a classical vacuum. The value of \( \mu_0 \) is approximately \( 4\pi \times 10^{-7} \) tesla meter per ampere (T·m/A).

The significance of \( \mu_0 \) is not only in its value but also in how it relates electric and magnetic field constants in the vacuum of space. In problems involving magnetic fields produced by currents, such as the magnetic field at a coil's center, it serves as a proportionality constant within the Biot-Savart Law and other electromagnetic equations.

When dealing with the magnetic fields of coils, students need to pay particular attention to the field at the center of the coil, which is of interest in various practical applications, including electromagnets and induction coils. For a circular coil, the magnetic field at its center is given by a simplified version of the Biot-Savart Law.

The formula for the magnetic field at the center of a circular coil carrying a current \(I\) is given by:
\( B = \frac{{\mu_0}}{{2}} * \frac{{NI}}{{r}} \) where \(N\) is the number of turns, and \(r\) is the radius of the coil. The reliance on \(N\) and \(r\) indicates that the field strength is directly proportional to the current and number of turns, and inversely proportional to the coil's radius. Knowing this helps in understanding why, for instance, adding more turns to a coil or increasing the current leads to a stronger magnetic field at its center.

In exercises such as the given problem, calculating the current in a circular coil involves rearranging the equation for the magnetic field at the coil's center to solve for the current \(I\). Using the given magnetic field \(B\), the number of turns \(N\), and the coil radius \(r\), we apply
\(I = \frac{{2Br}}{{\mu_0N}}\).

This formula indicates that the required current \(I\) is directly proportional to the desired magnetic field \(B\) and the radius of the coil \(r\), and inversely proportional to the permeability of free space \(\mu_0\) and the number of turns \(N\). Students can use this relationship to calculate the necessary current to produce a specific magnetic field at the center of a coil, which is a common exercise in both theoretical and practical contexts involving electromagnetism.