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When a loop carrying a current is placed in a magnetic field, it experiences a torque. This torque, denoted as \(\vec{\tau}\), is directly related to the magnetic moment of the loop. The concept of torque in magnetic fields can be understood through its formula:

• \(\vec{\tau} = \vec{\mu} \times \vec{B} \)

Here, \(\vec{\mu}\) is the magnetic moment, and \(\vec{B}\) is the magnetic field. The cross product implies that the torque direction is perpendicular to both the magnetic moment and the magnetic field.
The torque will try to align the magnetic moment with the magnetic field. This is analogous to the way a compass needle aligns with Earth's magnetic field, seeking a minimum energy state. In our scenario, the given torque \(D(4\hat{\imath} - 3\hat{\jmath})\) showcases how a current loop undergoes a rotational effect due to the magnetic field's components.

The magnetic field \(\vec{B}\) is characterized by its components in space: \(B_x\), \(B_y\), and \(B_z\). These components define the field's intensity in the \(x\), \(y\), and \(z\) directions respectively. Each component has a unique influence on the magnetic moment in terms of torque and potential energy.
The steps to find these components involve using the cross product from the torque equation:

• Given \(\vec{\tau} = D(4\hat{\imath} - 3\hat{\jmath})\)
• By relating it to \(-IA\hat{k} \times (B_x \hat{i} + B_y \hat{j} + B_z \hat{k})\), each component is determined.

Through solving the equations, we find:

• \(B_x = \frac{3D}{IA}\)
• \(B_y = \frac{4D}{IA}\)
• \(B_z = \frac{12D}{IA}\)

These components are essential to describe how the magnetic field influences both the loop's torque and alignment.

Magnetic potential energy relates to the orientation of a magnetic moment within a magnetic field. The potential energy \(U\) of a magnetic moment \(\vec{\mu}\) in a magnetic field \(\vec{B}\) is expressed as:

• \(U = - \vec{\mu} \cdot \vec{B} \)

Here, the negative sign indicates a desire for the system to reach lower energy states by aligning the moment with the field.
In our specific scenario, the negative potential energy implies that the loop is already near this stable orientation, despite resisting forces like torque that might seek to misalign it. The components \(B_x\), \(B_y\), and \(B_z\) contribute to this energy, dictating how strong the alignment tendency is in each spatial direction.
Understanding magnetic potential energy is vital for comprehending why objects like magnets and loops rotate within fields—they are seeking the lowest possible energy configuration.