91Ó°ÊÓ

The concept of a magnetic field is central to understanding how magnets and currents interact. In this exercise, a magnetic field is present, denoted by the vector \( \overrightarrow{\boldsymbol{B}} \), within which a circular loop carrying a current \( I \) is placed. The field exerts forces on moving charges or currents, which can lead to various phenomena such as torque.

A magnetic field can be visualized as lines extending from the north pole of a magnet to its south pole. These lines are denser where the field is stronger. The strength of the magnetic field is usually represented by the magnitude \( B \), and its direction by the orientation of the field lines.

In this exercise, we learn that the magnitude of the magnetic field is calculated using \( B_0 = \frac{13D}{IA} \). This tells us that the field strength is inversely proportional to the product of the current \( I \) and the area \( A \) of the loop.

Torque describes the twisting or rotational effect produced on an object. It is defined here by the relationship \( \overrightarrow{\tau} = \overrightarrow{\mu} \times \overrightarrow{B} \). This means the torque \( \overrightarrow{\tau} \) on the loop stems from the interaction of the magnetic moment \( \overrightarrow{\mu} \) with the magnetic field \( \overrightarrow{B} \).

In our problem, it’s given by \( \vec{\tau} = D(4\hat{i} - 3\hat{j}) \). The torque direction is significant because it hints at the relative orientations of \( \overrightarrow{\mu} \) and \( \overrightarrow{B} \).

The resulting torque is solely in the xy-plane, implying that the magnetic moment must lie in a direction (along the z-axis) perpendicular to this plane. Torque is larger when the magnetic moment is perpendicular to the field, as suggested by the cross product geometry.

Magnetic potential energy \( U \) is the energy stored due to an object's position in a magnetic field. It is described by \( U = -\overrightarrow{\mu} \cdot \overrightarrow{B} \). This formula highlights that the potential energy depends on the relative alignment of \( \overrightarrow{\mu} \) and \( \overrightarrow{B} \).

When \( U \) is negative, like in our exercise, it signifies that \( \overrightarrow{\mu} \) and \( \overrightarrow{B} \) face in the same general direction, with an angle less than 90 degrees between them.

Due to this configuration, the torque and potential energy conditions indicate the alignment and zero-out other components \( B_x \) and \( B_y \), emphasizing a simplification to focus on \( B_z \), hence \( \overrightarrow{B} = (0, 0, B_0) \). This alignment minimizes energy, stabilizing the physical system.