A magnetic dipole with a dipole moment of magnitude 0.020 J / T is released from rest in a unifo

step 1 So here is a magnetic dipole that has a dipole moment of magnitude 0 .02. So let me write that h

A magnetic dipole with a dipole moment of magnitude 0.020 J...

A magnetic dipole with a dipole moment of magnitude 0.020 $\mathrm{J} / \mathrm{T}$ is released from rest in a uniform magnetic field of magnitude 52 $\mathrm{mT}$ . The rotation of the dipole due to the magnetic force on it is unimpeded. When the dipole rotates through the orientation where its dipole moment is aligned with the magnetic field, its kinetic energy is 0.80 $\mathrm{mJ.}($ a) What is the initial angle between the dipole moment and the magnetic field? (b) What is the angle when the dipole is next (momentarily) at rest?

Key Concepts

Magnetic Dipole Moment

This is a vector quantity that represents the strength and orientation of a magnetic dipole. It quantifies how much a system will respond to an external magnetic field and is central to determining the magnetic potential energy and the torque experienced by the dipole in a field.

Uniform Magnetic Field

A uniform magnetic field is one in which the magnetic field strength and direction are constant in space. This concept is crucial because it simplifies the analysis of the forces and torques acting on a magnetic dipole, allowing the potential energy changes and rotational dynamics to be calculated using straightforward equations.

Magnetic Potential Energy

Magnetic potential energy in the context of a dipole in a magnetic field is given by the equation U = -m · B · cos?. It represents the energy associated with the orientation of a dipole relative to the magnetic field direction, with lower energy when the dipole aligns with the field and higher energy when it is opposed.

Torque on a Magnetic Dipole

A magnetic dipole in an external magnetic field experiences a torque given by ? = m × B, which tends to align the dipole with the field. This torque is key in analyzing the rotational motion of the dipole, as it dictates how the orientation changes over time.

Conservation of Energy in Rotational Dynamics

This principle states that, in the absence of non-conservative forces, the total mechanical energy of a system remains constant. For a rotating magnetic dipole, the loss in magnetic potential energy as it aligns with the field is converted into rotational kinetic energy, allowing the calculation of angular displacements based on energy conservation.

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