b. What is the loop's equilibrium orientation?

Verified step by step guidance

1. Identify the forces acting on the loop: In this scenario, the magnetic field exerts a torque on the current-carrying loop. The torque depends on the magnetic dipole moment of the loop and the external magnetic field.
2. Understand the magnetic dipole moment: The magnetic dipole moment ($\overrightarrow{\mu }$) of a current loop is given by $\overrightarrow{\mu }=I\cdot A\cdot \hat{n}$, where $I$ is the current, $A$ is the area of the loop, and $\hat{n}$ is the unit vector normal to the plane of the loop.
3. Calculate the torque on the loop: The torque ($\overrightarrow{\tau }$) exerted by the magnetic field on the loop is given by $\overrightarrow{\tau }=\overrightarrow{\mu }\times \overrightarrow{B}$, where $\overrightarrow{B}$ is the magnetic field.
4. Determine the equilibrium orientation: At equilibrium, the torque on the loop will be zero. This occurs when the magnetic dipole moment is aligned with the magnetic field, meaning $\overrightarrow{\mu }$ is parallel to $\overrightarrow{B}$.
5. Conclude the orientation: The loop's equilibrium orientation is such that its normal vector ($\hat{n}$) is aligned with the direction of the magnetic field.

Key Concepts

Equilibrium

Equilibrium in physics refers to a state where all forces and torques acting on an object are balanced, resulting in no net force or rotation. In the context of a loop, this means that the gravitational force, tension, and any other forces are in balance, allowing the loop to maintain a stable position without accelerating or rotating.

Torque

Torque is a measure of the rotational force applied to an object, which causes it to rotate around an axis. It is calculated as the product of the force applied and the distance from the pivot point to the line of action of the force. Understanding torque is essential for determining how forces affect the orientation of the loop in equilibrium.

Center of Mass

The center of mass is the point in an object where its mass is evenly distributed in all directions. For a loop, the center of mass plays a crucial role in determining its equilibrium orientation, as the loop will tend to orient itself such that the center of mass is at its lowest potential energy position, which is typically directly below the pivot point.