When a rigid charge distribution with charge Q and mass M...

When a rigid charge distribution with charge $Q$ and mass $M$ rotates about an axis, its magnetic moment $\vec{\mu}$ is linearly proportional to its angular momentum $\overrightarrow{\boldsymbol{L}},$ with $\overrightarrow{\boldsymbol{\mu}}=\gamma \overrightarrow{\boldsymbol{L}}$. The constant of proportionality $\gamma$ is called the gyromagnetic ratio of the object. We can write $\gamma=g(Q / 2 M),$ where $g$ is a dimensionless number called the $g$ -factor of the object. Consider a spherical shell with mass $M$ and uniformly distributed charge $Q$ centered on the origin $O$ and rotating about the $z$ -axis with angular speed $\omega$. (a) A thin slice with latitude $\theta$ measured with respect to the positive $z$ -axis describes a current loop with width $R d \theta$ and radius $r=R$ $\sin \theta,$ as shown in Fig. P28.77. Figure P28.77 What is the differential current $d I$ carried by this loop, in terms of $Q$. $\omega, R, \theta,$ and $d \theta ?$ (b) The differencontributed tial magnetic by that slice is $d \mu=A d I,$ where $A=\pi r^{2}$ is the area enclosed by the loop. Express the differential magnetic moment in terms of $Q$. $\omega, R, \theta,$ and $d \theta$ (c) Integrate over $\bar{\theta}$ to determine the magnetic moment $\vec{\mu}$. (d) What is the magnitude of the angular momentum $\overrightarrow{\boldsymbol{L}} ?$ (e) Determine the gyromagnetic ratio $\gamma .(\mathrm{f})$ What is the $g$ -factor for a spherical shell?

When a rigid charge distribution with charge $Q$ and mass $M$ rotates about an axis, its magnetic moment $\vec{\mu}$ is linearly proportional to its angular momentum $\overrightarrow{\boldsymbol{L}},$ with $\overrightarrow{\boldsymbol{\mu}}=\gamma \overrightarrow{\boldsymbol{L}}$. The constant of proportionality $\gamma$ is called the gyromagnetic ratio of the object. We can write $\gamma=g(Q / 2 M),$ where $g$ is a dimensionless number called the $g$ -factor of the object. Consider a spherical shell with mass $M$ and uniformly distributed charge $Q$ centered on the origin $O$ and rotating about the $z$ -axis with angular speed $\omega$. (a) A thin slice with latitude $\theta$ measured with respect to the positive $z$ -axis describes a current loop with width $R d \theta$ and radius $r=R$ $\sin \theta,$ as shown in Fig. P28.77. Figure P28.77 What is the differential current $d I$ carried by this loop, in terms of $Q$. $\omega, R, \theta,$ and $d \theta ?$ (b) The differencontributed tial magnetic by that slice is $d \mu=A d I,$ where $A=\pi r^{2}$ is the area enclosed by the loop. Express the differential magnetic moment in terms of $Q$. $\omega, R, \theta,$ and $d \theta$
(c) Integrate over
$\bar{\theta}$ to determine the magnetic moment $\vec{\mu}$. (d) What is the magnitude of the angular momentum $\overrightarrow{\boldsymbol{L}} ?$
(e) Determine the gyromagnetic ratio $\gamma .(\mathrm{f})$ What is the $g$ -factor for a spherical shell?

Key Concepts

Differential Contribution and Integration in Spherical Coordinates

To determine the overall magnetic moment of a rotating spherical shell, one breaks the shell into infinitesimal elements (or slices) and computes the contribution from each using spherical coordinates. By expressing the physical quantities (such as current and enclosed area) in terms of an angular variable and integrating over the entire range, one can sum these differential contributions to find total quantities. This method is widely used in physics to deal with continuous charge or mass distributions.

Spherical Shell Charge Distribution

A spherical shell with a uniformly distributed charge presents a symmetrical system that simplifies many electromagnetic calculations. In such a system, the rotational symmetry allows for the systematic calculation of physical quantities by integrating contributions from differential elements, making it a useful model for demonstrating principles like differential current and magnetic moment integration.

Rigid Body Rotation and Angular Momentum

In the context of a rotating charged body, the angular momentum is a natural consequence of its mass distribution and rotation. For a rigid body, the angular momentum is given by the product of its moment of inertia and angular velocity. Understanding this concept is essential when linking mechanical rotation to electromagnetic properties, one of the key links in the gyromagnetic studies.

g-factor

The g-factor is a dimensionless number that modifies the gyromagnetic ratio to account for the intrinsic properties of a system. It appears in the relationship ? = g(Q/2M), where Q and M are the charge and mass of the body, respectively. It serves as a correction factor that reflects the distribution of mass and charge, and deviations from classical expectations provide insights into the underlying physics.

Gyromagnetic Ratio

The gyromagnetic ratio is the proportionality constant between the magnetic moment and the angular momentum of a rotating charged object. It encapsulates how effectively angular momentum is converted into a magnetic moment. This concept bridges the study of magnetism and mechanics and is central when analyzing systems like electrons in atoms and macroscopic charged bodies.

Magnetic Moment of a Current Loop

The magnetic moment is a measure of the strength of a current loop's magnetic field. It is defined as the product of the current and the area of the loop (? = I A), and plays a crucial role in characterizing how current distributions produce magnetic fields. This concept is fundamental in electromagnetic theory, linking the motion of electric charges to magnetic effects.

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