Two long, parallel wires separated by a distance, \(d\), carry currents in opposite directions. If the left-hand wire carries a current \(i / 2,\) and the right-hand wire carries a current \(i\), determine where the magnetic field is zero.

The Biot-Savart Law is a critical principle in electromagnetism, specifically concerning the magnetic effects of electric currents. It provides a mathematical model for calculating the magnetic field produced at a point in space due to a current-carrying conductor. According to this law, the magnetic field \( B \) at a point is directly proportional to the current \( i \) and inversely proportional to the distance \( x \) from the conductor.

For a straight, infinite wire, the equation is expressed as:
\[ B = \frac{\mu_0 i}{2\pi x} \]
where \( \mu_0 \) is the permeability of free space, representing how much resistance the vacuum offers against the formation of the magnetic field. This law is indispensable when dealing with problems involving magnetic fields around currents, as evident in our textbook problem where it helps determine the magnetic field produced by two parallel wires carrying electric current.

Magnetic fields cancellation is a phenomenon where the magnetic fields from different sources combine to produce a net field of zero at certain points in space. This occurs when the magnetic fields are equal in magnitude but opposite in direction. In our exercise, we explored this concept by looking for a point where the magnetic fields produced by two currents in opposite directions cancel each other out.

Using the Biot-Savart Law, we find that at the point of cancellation, the field contributions from both wires must satisfy the condition:\[ B_1 = B_2 \]
By setting up and solving the equations, we determined the specific locations relative to the wires where this cancellation occurs. Understanding magnetic field cancellation is crucial for designing magnetic shielding and in applications that require neutral magnetic environments, such as in certain medical or scientific instrumentation.

The permeability of free space, denoted as \( \mu_0 \) and also known as the magnetic constant, is a fundamental physical constant that represents the measure of the magnetic permeability of the classical vacuum. Essentially, it's a measure of how much magnetic field is produced in a vacuum by an electric current or a magnetic moment. The value of \( \mu_0 \) is roughly \(4\pi \times 10^{-7} \, Tm/A\) (tesla meter per ampere).

In the equations derived from the Biot-Savart Law, \( \mu_0 \) appears as a multiplier, showing how it influences the strength of the magnetic field around a current-carrying wire. It's noteworthy that \( \mu_0 \) not only plays a role in this law but also in other electromagnetic phenomena, including the inductance of coils and the operation of electromagnetic waves.