Take a small rectangular current loop of sides \(\mathrm{d} x\), dy situated at the origin in the xy plane and carrying a current \(I\) round its four sides. Find the B-field at a. field point P a great distance from the loop along the \(x\) axis. Show that the field has the form \(\mu_{0} m /\left(4 \pi x^{3}\right)\), where \(m\) is its dipole moment. (Note that all four sides produce a field.)

The Biot-Savart Law is a foundational equation in electromagnetism that describes the magnetic field generated by a steady electric current. It reveals the relationship between the magnetic field \texttt{dB} at a point in space and a small current element that contributes to that field. Specifically, this law states that the magnetic field \texttt{dB} due to a segment of current \texttt{dl} is proportional to the current \texttt{I}, the length of the segment \texttt{dl}, the vector \texttt{r} connecting the segment to the point in space, and inversely proportional to the cube of the distance from the segment to the point. Mathematically, it is given by:
\[dB = \frac{\mu_0}{4\pi}\frac{I(dl\times r)}{r^{3}}\]
where \(\mu_0\) is the permeability of free space. The \texttt{dl×r} factor is a cross-product, indicating that the magnetic field is perpendicular to both the direction of the current and the line connecting the current to the point where the field is measured. The Biot-Savart Law is crucial for calculating the magnetic field produced by currents in different geometries, such as in the case of a current loop.

The magnetic dipole moment (\texttt{m}) is a vector quantity that represents the strength and orientation of a magnetic source, in this case, a current loop. It's related to the current \texttt{I} flowing through the loop and the area \texttt{A} the loop encloses, with \texttt{m} pointing perpendicular to the plane of the loop, following the right-hand rule. The dipole moment is a central concept as it helps quantify the torque a magnetic dipole experiences in an external magnetic field and the energy associated with its orientation in the field. For a rectangular loop, its magnetic dipole moment is given by:
\[m = I \cdot dx \cdot dy\]
where \texttt{dx} and \texttt{dy} are the lengths of the sides of the rectangular loop. This quantity simplifies the equations for magnetic fields at points far away from the loop, allowing us to express the field strength in terms of the dipole moment.

Magnetic field calculation involves determining the strength and direction of the magnetic field at a point in space, caused by electric currents. The magnetic field around a current loop can be calculated using the Biot-Savart Law. By integrating the contributions of magnetic field \texttt{dB} from all current elements around the loop and taking into consideration the symmetries and orientations, we can arrive at a simplified expression for the total magnetic field \texttt{B}. Notably, in this example, the field at a distant point P along the x-axis due to a small rectangular loop simplifies under the assumption that the point is very far compared to the loop size. This leads to a dipole field pattern where the field strength diminishes with the cube of the distance from the loop, following the relationship:
\[B = \frac{\mu_0 m}{4\pi x^3}\]
By invoking the concept of the magnetic dipole moment, the field calculation becomes more straightforward and illuminates the inverse cubic relationship between the magnetic field strength and the distance from the dipole.

The permeability of free space, denoted as \(\mu_0\), is a fundamental physical constant that describes how magnetic fields interact with the vacuum of empty space. It essentially characterizes the ability of the vacuum to support a magnetic field and is integral to magnetic field calculations in free space. In the context of the Biot-Savart Law, it helps determine the strength of the magnetic field produced by a current-carrying wire in free space. The value for the permeability of free space is roughly \(4\pi \times 10^{-7} \) Newtons per ampere squared (\texttt{N·A\textsuperscript{−2}}). It's a part of the Biot-Savart Law and appears in many equations of classical electromagnetism, such as that for the magnetic field around a current loop, which includes the factor \(\frac{\mu_0}{4\pi}\), representing the magnetic constant's role in dictating the magnitude of the magnetic interaction in a vacuum.