A square loop, with sides of length \(L\), carries current i. Find the magnitude of the magnetic field from the loop at the center of the loop, as a function of \(i\) and \(L\).

Ampere's Law is a foundational principle in electromagnetism that relates the magnetic field around a closed loop to the electric current passing through the loop. The law states that the line integral of the magnetic field \(\vec{B}\) along a closed path is proportional to the electrical current \(I_{enc}\) flowing through the enclosed area. The mathematical expression for Ampere's Law is:\
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\[\oint \vec{B} \cdot d\vec{l} = \mu_0 I_{enc}\]
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where \(\mu_0\) is the permeability of free space and \(d\vec{l}\) is an infinitesimal element of the closed path. When we use Ampere's Law to calculate the magnetic field at the center of a square loop, we essentially add up the magnetic effects of the current in each segment of the loop. Despite the complex appearance of the law, it provides a systematic approach for calculating magnetic fields, especially in symmetric configurations, such as loops and coils.

Calculating the magnetic field produced by a current-carrying conductor involves understanding the geometry of the conductor and the position where the field is being calculated. In the case of a square loop, each side contributes a magnetic field component at the center of the loop. By exploiting symmetry and using laws like Ampere's, we can determine that the resultant magnetic field is perpendicular to the plane of the loop.
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In the case of the square loop, the problem simplifies due to each side producing an identical magnetic field at the center. This allows us to calculate the field for one side and then multiply by the number of sides. The methodical step-by-step solution demonstrated how we harness Ampere's Law to compute \(B_\perp\) for one side and then arrive at the total magnetic field \(B_{total}\) for the whole loop by multiplying by four.

The permeability of free space, denoted by \(\mu_0\), plays a crucial role in the relationship between electric currents and magnetic fields. This constant represents the extent to which vacuum allows the formation of a magnetic field. Its value is approximately \(4\pi \times 10^{-7} \text{T}·\text{m/A}\), which means that it relates the magnetic field in teslas (T) to the current in amperes (A) and the distance in meters (m).
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In the solution provided for the square loop, \(\mu_0\) is integral to the final expression for the magnetic field. It is the proportionality constant that makes the relationship precise, allowing for a magnetic field calculation that can be directly tied to the physical characteristics of the loop—the current \(i\) and the side length \(L\).

One of the fundamental tenets of electromagnetism is that electric currents produce magnetic fields. This relationship is exploited when calculating the magnetic field from a current-carrying loop. The direction of the magnetic field produced by a straight current-carrying wire is given by the right-hand rule, and its magnitude is proportional to the current and inversely proportional to the distance from the wire.
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The square loop example showcases this relationship. Each side of the loop is considered a segment of current that generates a magnetic field at the loop’s center. By calculating the field produced by each segment and considering the direction and proportionality, we obtain the total field resulting from the entire current. The cumulative magnetic field in the center of the loop shows a linear relationship with the current \(i\) and an inverse relationship with the loop side length \(L\).