**Title: Analysis of Electric Fields in a Coaxial Cylindrical Cable with Dielectric Material**A long coaxial cylindrical cable consists of a metal wire of radius \( a \), which is surrounded by a concentric metal tube of inner radius \( b \). The space between them is filled with a linear dielectric material of susceptibility \( \chi \). A free charge per unit length \( \lambda = Q/L \) is placed on the inner cable, and an opposite charge per unit length is placed on the outer tube.### Questions:**(a)** Using Gauss’s law, find the electric displacement \( D \) in the region between the two conductors \( a < s < b \), where \( s \) is the distance from the cylindrical axis. [Details on the application of Gauss's law and calculations needed]**(b)** Using the susceptibility \( \chi \), find the electric field \( E \) and the polarization \( P \) in the region between the two conductors. [Specific steps and formulae for calculating \( E \) and \( P \)]**(c)** Find the bound volume and surface charge densities on the dielectric material. [Explanation of concepts and equations used]**(d)** Verify that the discontinuity of \( E \) at \( s = a \) satisfies Gauss’s law for the total electric charge. [Verification steps and calculations]---### Explanation:- **Electric Displacement (\( D \))**: Represents how electric field influences the organization of charges in the dielectric material. - **Electric Field (\( E \))**: A vector field representing the force exerted by the electric charges. - **Polarization (\( P \))**: The vector field that expresses the density of electric dipole moments in a dielectric material.- **Bound Charge Densities**: Result from the polarization of the dielectric material when exposed to an electric field.This exercise will enhance your understanding of electromagnetic theory in cylindrical coordinates and the behavior of dielectrics in electric fields.

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A long coaxial cylindrical cable consists of a metal wire of radius a, which is surrounded by a concentric metal tube of inner radius b. The space between them is filled with a linear dielectric material of susceptibility x. A free charge per length 1 = Q/L is placed on the inner cable and an opposite charge per length is placed on the outer tube. (a) Using Gauss's law find the electric displacement D in the region between the two conductors a

A long coaxial cylindrical cable consists of a metal wire of radius a, which is surrounded by a concentric metal tube of inner radius b. The space between them is filled with a linear dielectric material of susceptibility x. A free charge per length 1 = Q/L is placed on the inner cable and an opposite charge per length is placed on the outer tube. (a) Using Gauss's law find the electric displacement D in the region between the two conductors a

Introductory Circuit Analysis (13th Edition)

13th Edition

ISBN:9780133923605

Author:Robert L. Boylestad

Publisher:Robert L. Boylestad

Chapter1: Introduction

Section: Chapter Questions

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Concept explainers

Electric field

A physical field present around an electrically charged particle that exerts a force on all other charged particles is called an electric field. The charged particles present in the field either repel or attract. From electric charges or time-varying magnetic fields, the electric field originates. One of the manifestations of electromagnetic force which is one of the four fundamental forces present in nature are the electric fields and magnetic fields.

Question

**Title: Analysis of Electric Fields in a Coaxial Cylindrical Cable with Dielectric Material**

A long coaxial cylindrical cable consists of a metal wire of radius \( a \), which is surrounded by a concentric metal tube of inner radius \( b \). The space between them is filled with a linear dielectric material of susceptibility \( \chi \). A free charge per unit length \( \lambda = Q/L \) is placed on the inner cable, and an opposite charge per unit length is placed on the outer tube.

### Questions:

**(a)** Using Gauss’s law, find the electric displacement \( D \) in the region between the two conductors \( a < s < b \), where \( s \) is the distance from the cylindrical axis. [Details on the application of Gauss's law and calculations needed]

**(b)** Using the susceptibility \( \chi \), find the electric field \( E \) and the polarization \( P \) in the region between the two conductors. [Specific steps and formulae for calculating \( E \) and \( P \)]

**(c)** Find the bound volume and surface charge densities on the dielectric material. [Explanation of concepts and equations used]

**(d)** Verify that the discontinuity of \( E \) at \( s = a \) satisfies Gauss’s law for the total electric charge. [Verification steps and calculations]

---

### Explanation:
- **Electric Displacement (\( D \))**: Represents how electric field influences the organization of charges in the dielectric material.

- **Electric Field (\( E \))**: A vector field representing the force exerted by the electric charges.

- **Polarization (\( P \))**: The vector field that expresses the density of electric dipole moments in a dielectric material.

- **Bound Charge Densities**: Result from the polarization of the dielectric material when exposed to an electric field.

This exercise will enhance your understanding of electromagnetic theory in cylindrical coordinates and the behavior of dielectrics in electric fields.

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