In Lecture 9, we explored that the energy density of an electric field is given by \( u = \varepsilon_0 E^2 /2 \). This "electric-field energy" is not a new form of energy; rather, it is a new way to interpret electric potential energy. Particularly, we can calculate the total potential energy of a charge distribution using the energy density.Previously, we computed the total potential energy for a collection of point charges (see slide 13 in Lecture 6). Now, we can calculate it for continuous charge distributions. The total potential energy of a charge distribution is the volume integral of the energy density across all regions of space:\[U = \int_{\text{all space}} \frac{1}{2} \varepsilon_0 E^2 \, dV\](a) Consider a positive charge \( Q \) distributed uniformly across the surface of a thin spherical shell with radius \( R \): i. Determine the electric field in all regions of space. ii. Use the integral above to calculate the total potential energy of this charge distribution.(b) Consider a positive charge \( Q \) distributed uniformly throughout the volume of a solid sphere of radius \( R \): i. Determine the electric field in all regions of space. ii. Use the integral above to calculate the total potential energy of this charge distribution.(Note: "all regions of space" truly includes all regions. For both cases, you need to perform integration over the volumes inside and outside the sphere.)

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J2 all space (a) Positive charge Q is distributed uniformly over the surface of a thin spherical shell of radius i. ii. Find the electric field in all regions of space. Calculate the total potential energy of this charge distribution using the integral ab (b) Positive charge Q is distributed uniformly throughout the volume of a solid sphere of radiu: Find the electric field in all regions of space. i. ii. Calculate the total potential energy of this charge distribution using the integral ab (Note: "all regions of space" really does mean all regions. In both cases, you need to integrate the volumes inside and outside the sphere.)

J2 all space (a) Positive charge Q is distributed uniformly over the surface of a thin spherical shell of radius i. ii. Find the electric field in all regions of space. Calculate the total potential energy of this charge distribution using the integral ab (b) Positive charge Q is distributed uniformly throughout the volume of a solid sphere of radiu: Find the electric field in all regions of space. i. ii. Calculate the total potential energy of this charge distribution using the integral ab (Note: "all regions of space" really does mean all regions. In both cases, you need to integrate the volumes inside and outside the sphere.)

Introductory Circuit Analysis (13th Edition)

13th Edition

ISBN:9780133923605

Author:Robert L. Boylestad

Publisher:Robert L. Boylestad

Chapter1: Introduction

Section: Chapter Questions

Problem 1P: Visit your local library (at school or home) and describe the extent to which it provides literature...

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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

In Lecture 9, we explored that the energy density of an electric field is given by \( u = \varepsilon_0 E^2 /2 \). This "electric-field energy" is not a new form of energy; rather, it is a new way to interpret electric potential energy. Particularly, we can calculate the total potential energy of a charge distribution using the energy density.

Previously, we computed the total potential energy for a collection of point charges (see slide 13 in Lecture 6). Now, we can calculate it for continuous charge distributions. The total potential energy of a charge distribution is the volume integral of the energy density across all regions of space:

\[
U = \int_{\text{all space}} \frac{1}{2} \varepsilon_0 E^2 \, dV
\]

(a) Consider a positive charge \( Q \) distributed uniformly across the surface of a thin spherical shell with radius \( R \):

i. Determine the electric field in all regions of space.

ii. Use the integral above to calculate the total potential energy of this charge distribution.

(b) Consider a positive charge \( Q \) distributed uniformly throughout the volume of a solid sphere of radius \( R \):

i. Determine the electric field in all regions of space.

ii. Use the integral above to calculate the total potential energy of this charge distribution.

(Note: "all regions of space" truly includes all regions. For both cases, you need to perform integration over the volumes inside and outside the sphere.)

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