A circular loop of wire with area A lies in the x y -plane....

A circular loop of wire with area $A$ lies in the $x y$ -plane. As viewed along the $z$ -axis looking in the $-z$ -direction toward the origin, a current $I$ is circulating clockwise around the loop. The torque produced by an extemal magnetic field $\vec{B}$ is given by $\vec{\tau}=D(4 \hat{z}-3 \hat{y}),$ where $D$ is a positive constant, and for this orientation of the loop the magnetic potential energy $U=-\vec{\mu} \cdot \vec{B}$ is negative. The magnitude of the magnetic field is $B_{0}=13 D / L A$ (a) Determine the vector magnetic moment of the current loop. (b) Determine the components $B_{x}, B_{y}$ , and $B_{z}$ of $\vec{B}$ .

A circular loop of wire with area $A$ lies in the $x y$ -plane. As viewed along the $z$ -axis looking in the $-z$ -direction toward the origin, a current $I$ is circulating clockwise around the loop. The
torque produced by an extemal magnetic field $\vec{B}$ is given by $\vec{\tau}=D(4 \hat{z}-3 \hat{y}),$ where $D$ is a positive constant, and for this orientation of the loop the magnetic potential energy $U=-\vec{\mu} \cdot \vec{B}$ is negative. The magnitude of the magnetic field is $B_{0}=13 D / L A$
(a) Determine the vector magnetic moment of the current loop.
(b) Determine the components $B_{x}, B_{y}$ , and $B_{z}$ of $\vec{B}$ .

Key Concepts

Magnetic Dipole Moment

The magnetic dipole moment is a vector quantity that characterizes the overall magnetic strength and orientation of a current loop or magnet. For a current-carrying loop, it is defined as the product of the current and the area of the loop, with the direction given by the right-hand rule. This concept is fundamental for analyzing the magnetic properties of systems and formulating their interactions with external magnetic fields.

Torque on a Magnetic Dipole

The torque experienced by a magnetic dipole in an external magnetic field is given by the cross product of the magnetic moment and the magnetic field. This torque tends to align the dipole with the field, and its magnitude depends on the sine of the angle between the magnetic moment and the magnetic field vectors. This concept is essential in understanding how magnetic moments respond to applied magnetic fields in various physical systems.

Magnetic Potential Energy

Magnetic potential energy describes the energy stored in a magnetic dipole when it is placed in an external magnetic field. This energy is given by the negative dot product of the magnetic dipole moment and the magnetic field, indicating that the energy is minimized when the dipole is aligned with the field. This concept plays a key role in the stability analysis and equilibrium orientations of magnetic systems.

Vector Decomposition of Magnetic Field

Determining the components of a magnetic field involves breaking down the vector field into its constituent parts along the coordinate axes. This process is crucial when the response of a magnetic dipole, such as the torque, is provided in vector form, requiring an analysis of its directional components. Understanding vector decomposition aids in the precise characterization and manipulation of magnetic fields in complex geometries.

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