A simplified model of Earth's magnetic field has it originating in a single current loop at the outer edge of the planet's liquid core (radius \(3000 \mathrm{km}\) ). What current would give the \(62-\mu \mathrm{T}\) field measured at the north magnetic pole?

The Biot-Savart Law is a fundamental principle in electromagnetism, providing a way to calculate the magnetic field generated by a steady current. This law states that the magnetic field \( B \) at a point in space is directly proportional to the current \( I \) and inversely proportional to the distance from the current-carrying wire or loop. The equation for the magnetic field created by a small segment of current-carrying wire is given as:
\[ d\textbf{B} = \frac{{\textbf{j} \times d\textbf{l} }}{{R^2}} \times \frac{{\textbf{μ_0}}}{{4\pi}} \]
where \( d\textbf{B} \) is the infinitesimal magnetic field produced by the current element \( d\textbf{l} \) carrying current density \( \textbf{j} \) at a distance \( R \) from the point of interest. The \( \textbf{μ_0} \) represents the permeability of free space, a physical constant expressing the ability of the vacuum to support magnetic fields.
The Biot-Savart Law is essential for understanding not only man-made electrical devices but also natural phenomena such as Earth's magnetic field. The law can be applied to calculate the magnetic field of relatively simple configurations of electric current, such as a straight wire, a solenoid, or a loop, which is the basis for solving the exercise given.

Understanding how to calculate a magnetic field is crucial in various fields of study, such as physics, engineering, and even environmental sciences when considering the Earth's magnetosphere. The magnetic field calculation involves inputting known values into the appropriate magnetic field formula derived from the Biot-Savart Law or Ampere's Law for different configurations of currents.
For a circular loop of wire, like in the textbook exercise, the formula for the magnetic field at the center of the loop simplifies to:
\[ B = \frac{μ_0 I}{2 R} \]
where \( B \) is the magnetic field, \( I \) is the current through the loop, and \( R \) represents the radius of the loop. This formula gives insight into how current and the geometry of the current's path affect the produced magnetic field.
By rearranging this formula, as seen in Step 2 of the solution provided, it enables solving for the unknown current \( I \) when the magnetic field \( B \) and radius \( R \) are known. This type of calculation is not only useful in academic exercises but plays a significant role in designing electrical circuits, magnetic field sensors, and analyzing geophysical data.

The permeability of free space, symbolized as \( \textbf{μ}_0 \) and sometimes called the magnetic constant, is a physical constant that represents the measure of the ability of the vacuum to sustain magnetic fields. In the International System of Units (SI), it is precisely defined to be \( 4\pi \times 10^{-7} \text{T}·\text{m/A} \) (tesla meter per ampere).
This constant is a cornerstone in the equations for the Biot-Savart Law as well as Ampere’s Law and is used when calculating the magnetic field produced by currents in free space. \( \textbf{μ}_0 \) also enters the definition of other physical constants such as the fine-structure constant and the impedance of free space.
In the context of the exercise involving the Earth's magnetic field, the permeability of free space is used to connect the measured magnetic field strength to the required current that would produce such a field. It helps us understand how the space through which the magnetic field permeates, even though considered a vacuum, plays a role in the transmission of magnetic effects.