A cylindrical shell with radius R and length W carries a...

A cylindrical shell with radius $R$ and length $W$ carries a uniform charge $Q$ and rotates about its axis with angular speed $\omega .$ The center of the cylinder lies at the origin $O$ and its axis is coincident with the $x$ -axis, as shown in Fig. P28.76. (a) What is the charge density $\sigma$ ? (b) What is the differential current $d I$ on a circular strip of the cylinder centered at $x$ and with width $d x$ ? (c) Use Eq. (28.15) to write an expression for the differential magnetic field $d \vec{B}$ at the origin due to this strip. (d) Integrate to determine the magnetic field at the origin.

A cylindrical shell with radius $R$ and length $W$ carries a uniform charge $Q$ and rotates about its axis with angular speed $\omega .$ The center of the cylinder lies at the origin $O$ and its axis is coincident with the $x$ -axis, as shown in Fig. P28.76.
(a) What is the charge density $\sigma$ ?
(b) What is the differential current $d I$ on a circular strip of the cylinder centered at $x$ and with width $d x$ ? (c) Use Eq. (28.15) to write an expression for the differential magnetic field $d \vec{B}$ at the origin due to this strip. (d) Integrate to determine the magnetic field at the origin.

Key Concepts

Integration over Extended Sources

Integration is a mathematical process used to sum the contributions of infinitesimal elements over a continuous distribution. In the context of electromagnetic fields, integration allows us to calculate the net field produced by an object whose properties vary continuously in space. This approach is essential when dealing with distributed charges or currents, where the overall effect is the accumulation of local effects computed via differential elements.

Differential Current Element in Extended Systems

In systems where the charge distribution is continuous and extended, it is often useful to consider infinitesimal elements of the charged body. Each differential element contributes a small amount of current, which can be calculated using the local charge density and velocity. The summation or integration over these differential contributions yields the total current distribution, which is necessary to analyze the resulting magnetic field.

Biot-Savart Law

The Biot-Savart law is a fundamental principle used to calculate the magnetic field produced by a current-carrying element. It relates the magnetic field contribution at a point in space to the current element and the geometry of the setup. In problems involving extended charge distributions, the law is applied to differential elements, followed by integration over the entire distribution to obtain the net magnetic field.

Current due to Rotating Charge

When a charged body rotates, its motion gives rise to a current due to the moving charge. This concept transforms a static charge distribution into a dynamic one, effectively creating a current loop or series of current elements. Understanding this conversion is crucial for calculating magnetic fields generated by rotating charged objects and interpreting how the rotation parameters such as angular speed affect the magnitude of the induced currents.

Surface Charge Density

This concept involves the distribution of electric charge uniformly over a surface. In problems involving a cylindrical shell or other geometries, the surface charge density is defined as the total charge divided by the total area over which the charge is spread. It is fundamental in relating the total charge present to local properties of the electrostatic field emanating from or interacting with the surface.

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