Find an expression for the magnetic field at the center of a square loop of side \(a\) carrying current \(I\)

When studying electromagnetism, the Biot–Savart Law is a critical equation that helps us understand how currents produce magnetic fields. Simply put, it states that the magnetic field (\r dB\r) created by a small segment of current is proportional to the size of the current (\r I\r), the length of the current element (\r dL\r), and is inversely proportional to the square of the distance (\r r\r) from the current element to the point where the magnetic field is being measured.

This law is mathematically expressed as \r dB = \frac{\mu_0I \r dL \times \hat{r}}{4\pi r^2}\r, where \(\mu_0\) represents the permeability of free space and \(\hat{r}\) is the unit vector pointing from the current element to the point of interest. To find the total magnetic field, one would integrate the contributions from all segments of the current-carrying wire or loop. This is what's done in Step 2 of the provided solution for a side of the square loop.

The permeability of free space, denoted as \(\mu_0\), is a fundamental physical constant that characterizes how a magnetic field can permeate the vacuum of space. Essentially, it is a measure of the magnetic responsiveness of the classical vacuum to the magnetic field. Its value is \(4\pi \times 10^{-7} \) newton per ampere squared (N/A²).

When calculating the magnetic field using the Biot–Savart Law, \(\mu_0\) is the proportionality constant, ensuring that the units and magnitudes are consistent. In the exercise solution, \(\mu_0\) is used to determine the strength of the magnetic field created by the current in the square loop. An analogy would be to imagine \(\mu_0\) as an 'easiness' factor for magnetic field lines to penetrate the surrounding space.

The right-hand thumb rule is a mnemonic that helps predict the direction of the magnetic field generated by a current-carrying conductor. To use this rule, extend your right thumb and curl the fingers. If you align your thumb in the direction of current flow through a wire, your fingers will curl in the direction of the magnetic field lines encircling the wire.

During Step 1 of the solution process, the right-hand thumb rule is utilized to establish that the magnetic fields induced by each side of the square loop all point in the same direction—perpendicular to the plane of the loop. This rule is an essential tool for visually grasping the directional aspect of electromagnetic interactions, and it simplifies complex three-dimensional field analyses to a simple hand gesture.