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The Biot-Savart Law is essential to understanding how magnetic fields are created by moving electric charges, such as those in a current-carrying wire. It states that at any point in space, the magnetic field (\textbf{dB}) generated by an infinitesimal segment of current (\textbf{Idl}) is directly proportional to the current and the length of the segment, and inversely proportional to the square of the distance (\textbf{r}) from the segment to the point.

The law can be mathematically expressed as: \[dB = \frac{\mu_0}{4\pi} \frac{I(dl \times \hat{r})}{r^2}\]
Here, \(\mu_0\) represents the permeability of free space, \(\mathbf{dl}\) is a vector representing the direction and magnitude of the current element, \(\hat{r}\) is a unit vector pointing from the current element to the point of interest, and \(r\) is the distance to the point at which the magnetic field is being calculated.

The significance of the Biot-Savart Law lies in its ability to calculate the magnetic field produced by any current configuration by integrating this law over the current path. This law forms the basis for designing electromagnets, electric motors, and understanding the magnetic effects of electrical currents in various applications.

Calculating the magnetic field around a current-carrying wire is a fundamental application of the Biot-Savart Law. For a long, straight wire, the calculation is simplified by the symmetrical distribution of the magnetic field lines. The field at a distance \(r\) from a wire carrying current \(I\) can be found using the simplification of the Biot-Savart Law:

\[B = \frac{\mu_0 I}{2\pi r}\]
This formula tells us the strength of the magnetic field a certain distance away from a straight wire. Key factors to note are: the magnetic field is directly proportional to the current \(I\), and it diminishes with distance \(r\) from the wire.

When dealing with multiple wires, as in the exercise provided, each wire creates its own magnetic field, and the total field at any given point is the vector sum of the individual fields. This concept is important for understanding complex systems with multiple current paths and how they interact to create the magnetic environment around them.

The magnetic field around a current-carrying wire not only demonstrates a fundamental electromagnetic phenomenon but also has practical implications in technology. The magnetic effects of such a wire include electromagnetic force on nearby magnets or other current-carrying wires, the basis for electrical measuring instruments, and the principle behind the operation of electric motors.

For two parallel wires carrying currents in the same direction, as in the original exercise, they will experience an attractive force, because the magnetic field produced by one wire will exert a force on the other wire. When the magnetic fields from different wires combine at a point, the resulting field can either increase or decrease depending on the direction of the currents. If the current direction is opposite in the two wires, they will repel each other.

The workings of an electric motor are based on these principles, where a current-carrying loop is placed in a magnetic field, and the interaction between the field produced by the loop and the external field results in rotational motion. Hence, the study of current-carrying wires and the magnetic fields they produce is not only of theoretical importance but is crucial in the design and functioning of many electronic devices and systems.