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The Biot-Savart Law is a fundamental principle in electromagnetism that describes how currents produce magnetic fields. A current-carrying wire generates a magnetic field around it, and the Biot-Savart Law gives us the means to calculate the magnitude and direction of this field at any point in space. According to this law, the differential magnetic field, denoted as \(dB\), produced by a small segment of current \(dI\) is directly proportional to the current, the length of the small segment \(dL\), and the sine of the angle between \(dL\) and the position vector \(r\), and inversely proportional to the square of the distance from the segment to the point where the field is being calculated.

The mathematical expression of the Biot-Savart Law is: \[dB = \frac{\rm{\boldsymbol{\beta}}_{0}I}{4\rm{\boldsymbol{\beta}}}\frac{dL \times r}{r^{3}}\] Here, \(dB\) is in the direction perpendicular to the plane formed by \(dL\) and \(r\), following the right-hand rule. This law is vital for calculating the magnetic field contributions from different segments of a wire or any other shape carrying current.

When calculating the magnetic field due to a current-carrying wire, it's crucial to consider the contributions from each part or segment of the wire. For a wire bent at a right angle, the magnetic field at a point in space is the vector sum of the fields produced by the vertical and horizontal sections of the wire. For the right-angled wire in the exercise, we needed to calculate the magnetic field contributions separately from the vertical and the horizontal segments.

Applying the Biot-Savart Law to the vertical segment, which runs parallel to the \(y-axis\), we find a contribution to the magnetic field. Similarly, for the horizontal segment along the \(x-axis\), another component of the field is determined. By vectorially adding these contributions, we obtain the total magnetic field at a point in the \(xy\) plane. This concept demonstrates the principle of superposition, where the total field is the sum of all individual field contributions.

In the context of magnetic fields generated by currents, the vector cross product is a crucial mathematical operation. It's used to calculate the direction and magnitude of the magnetic field produced by a current element. When applying the Biot-Savart Law, the differential magnetic field \(dB\) is given by the cross product of the current element \(dL\) and the position vector \(r\).

This cross product is denoted by \(dL \times r\) and results in a vector that is perpendicular to both \(dL\) and \(r\), obeying the right-hand rule. For instance, if \(dL\) is pointing in the direction of the current and \(r\) is pointing from the wire to the point where we calculate the field, then \(dB\) will be directed into or out of the plane, depending on the orientation of \(dL\) and \(r\). This is why, in the given exercise, magnetic field contributions from the vertical and horizontal sections had different signs, reflecting their different directions into the plane of the paper.

Coordinate transformation is the process of changing from one coordinate system to another. In electrodynamics, this is often used to simplify the mathematical calculation of fields. As seen in the exercise, transforming Cartesian coordinates \((x, y)\) to polar coordinates \((r, \theta)\) aids in integrating the contributions to the magnetic field from different parts of a wire or a loop.

For a point \(P\) in the \(xy\) plane, we can express its position in terms of \(r\) and \(\theta\) through the transformations \(x = r\rm{\boldsymbol{\beta}}\rm{\boldsymbol{\beta}}\rm{\boldsymbol{\beta}} \theta\) and \(y = r\rm{\boldsymbol{\beta}}\rm{\boldsymbol{\beta}}\theta\). These transformations facilitate determining the increments \(dx\) and \(dy\) used in integrating the magnetic field along the wire. By using coordinate transformation efficiently, complex problems involving the magnetic field of current-carrying wires can be made more tractable.