91Ó°ÊÓ

Determine the magnetic moment \(\overrightarrow{\boldsymbol{\mu}}\) of a spherical shell with radius \(R\) and uniform charge \(Q\) rotating with angular speed \(\vec{\Omega}=\omega \hat{k} .\) Use the following steps: (a) Consider a coordinate system with the origin at the center of the sphere. Parameterize each latitude on the sphere with the angle \(\theta\) measured from the positive \(z\) -axis. There is a circular current loop at each value of \(\theta\) for \(0 \leq \theta \leq \pi .\) What is the radius of the loop at latitude \(\theta ?\) (b) The differential current carried by that loop is \(d I=\sigma v d W,\) where \(\sigma\) is the charge density of the sphere, \(v\) is the tangential speed of the loop, and \(d W=R d \theta\) is its differential width. Express \(d I\) in terms of \(\sigma, R, \omega, \theta,\) and \(d \theta .\) (c) The differential magnetic moment of the loop is \(d \mu=A d I,\) where \(A\) is the area enclosed by the loop. Express \(d \mu\) in terms of \(R, \omega, \theta,\) and \(d \theta\) (d) Integrate over the sphere to determine the magnetic moment. Express your result as a vector; use the total charge \(Q\) rather than the charge density \(\sigma\). (e) If the sphere is in a uniform magnetic field \(\overrightarrow{\boldsymbol{B}}=(\sin \alpha \hat{\imath}+\cos \alpha \hat{\jmath}) B,\) what is the torque on the sphere?

Step by step solution

01

(a) Determine the Radius of the Loop at Latitude θ

Considering a spherical coordinate system with the origin at the center of the sphere, we measure the angle θ from the positive z-axis. The radius \(r\) of the loop at latitude \(θ\) can be obtained from the relationship \(r = R \sin θ \). This relationship comes from considering a right triangle formed by the radius R, the loop’s radius r, and the displacement from the z-axis.

02

(b) Express the Differential Current dI

The differential current carried by the loop is \(dI = σv dW \), where \(σ\) is the charge density of the sphere, \(v\) is the tangential speed of the loop (given by \(Rω\sinθ\), where \(ω\) is the angular speed), and \(dW\) is the differential width \(Rdθ\). Therefore, \( dI = σRω\sin(\theta) dW = σωR^2\sinθdθ \).

03

(c) Determine the Differential Magnetic Moment dμ

The differential magnetic moment of the loop is \( dμ = AdI \), where \(A\) is the area enclosed by the loop. Since the loop is a circle of radius \( R\sinθ \), the area \( A = π(R\sinθ)^2 = πR^2\sin^2θ \). Therefore, \(dμ = AdI = πωσR^4\sin^3θdθ\).

04

(d) Integrate to Determine the Magnetic Moment

Integrating the infinitesimal magnetic moments \(dμ = πωσR^4\sin^3θdθ\) over the sphere will determine the magnetic moment. The limits of the integral are from \(θ = 0\) to \(θ = π\), implying an integration over the entire sphere. Σμ = ∫dμ = ∫[πωσR^4\sins^3θdθ] from 0 to π. Which gives \( μ = πωσR^4 /4.\) To express it in terms of Q (the total charge of the sphere), knowing that \(Q=4πR^2σ\) (the total charge is the charge density times the surface area), we find \( μ = QωR^2 /16 \) in the z-direction.

05

(e) Determine the Torque on the Sphere

The torque on a magnetic body in a magnetic field is given by the cross product \(\vec{τ} = \vec{μ} × \vec{B}\). Here, since \(\vec{μ}\) is in the z-direction and \(\vec{B} = (\sin α \hat{i} + \cos α \hat{j})B\), taking the cross product provides the torque: \( \vec{τ} = QωR^2B \sinα /16 \hat{j} - QωR^2B \cosα /16 \hat{i} \).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Spherical Shell

A spherical shell is a three-dimensional hollow structure where the outer surface forms a sphere. In our context, the spherical shell possesses a uniform charge that is distributed over its surface. Imagine the surface of a basketball; all the air inside is hollow.

This visualization helps us grasp how charges are distributed over the shell. Since we are dealing with a shell, not a solid sphere, it’s crucial to focus on the two-dimensional surface of the sphere.

• The shell has a defined radius, often denoted as \(R\).
• It carries a total charge \(Q\) that is evenly spread across the surface.

Understanding this concept is key to exploring characteristics like its charge distribution and how its rotation influences other properties, such as its magnetic moment.

Angular Speed

Angular speed, denoted as \(\omega\), measures how fast an object rotates. Picture it as the rotational counterpart to linear speed.

When dealing with a spherical shell like the one in our problem, it’s rotating around an axis, like the Earth around its poles. This speed impacts how charge on the surface moves.

Here are a few key points about angular speed:

• It is typically expressed in radians per second.
• In a rotating system, all points along a radius from the rotation axis have the same angular speed.
• For the spherical shell, this rotation involves the entire structure, affecting the movement of charged particles on its surface.

Understanding angular speed is crucial when analyzing rotational motion and its effects on magnetic properties, as charged particles in motion create magnetic fields.

Charge Density

Charge density \(\sigma\) describes how charge \(Q\) is distributed over a surface or volume. In our exercise, we focus on the surface charge density, which measures the amount of charge per unit area on the sphere's surface.

Think of how grains of sand spread over a surface; charge density works similarly, signifying concentration in a given space.

• For a spherical shell, \(\sigma\) is calculated as total charge \(Q\) divided by the sphere's surface area \(4\pi R^2 \).
• Knowing \(\sigma\) aids in understanding current flow when the shell rotates, as it gives the charge available to move at each point.

Recognizing the role of charge density is essential to comprehending how the shell's rotation induces a magnetic moment, by seeing it as a collection of small current loops each contributing to the total magnetic field.

Current Loop

A current loop is essentially a path along which electric current flows, forming a closed loop. In the context of the spherical shell, each latitude line—like equators or parallels on a globe—acts as a loop when the sphere rotates.

As the spherical shell spins, charges move along these loops, creating currents due to the shell's angular speed. Let's look more closely:

• Each latitude represents a "circle" on the sphere's surface where charge moves.
• The radius of each current loop changes with the latitude \(\theta\), calculated as \(R \sin \theta\).
• The moving charge creates a magnetic field, and each loop contributes to the shell's overall magnetic moment.

Understanding how these current loops work is key to solving our problem, as they reveal how the shell's rotation links to the generated magnetic fields. Thus, these loops help define the resultant magnetic moment of the entire shell.