Physics for Scientists and Engineers
Physics for Scientists and Engineers
6th Edition
ISBN: 9781429281843
Author: Tipler
Publisher: MAC HIGHER
bartleby

Concept explainers

bartleby

Videos

Question
Book Icon
Chapter 22, Problem 86P

a.

To determine

To Show:The net electric flux of the electric field out of the Gaussian surface is given by ϕnetδExδxΔV

The net electric flux of the electric field out of the Gaussian surface is given by ϕnetδExδxΔV

Given:

A small gaussian surface as shown

  Physics for Scientists and Engineers, Chapter 22, Problem 86P , additional homework tip  1

Formula Used:

The net flux is given by

  ϕnet=ϕ(x+Δx)ϕ(x)

Calculations:

The net flux is given by

  ϕnet=ϕ(x+Δx)ϕ(x)

Using Taylor series expansion

  ϕnet=ϕ(x)+(Δx)ϕ'(x)+1/2ϕ''(x)+...ϕ(x)=(Δx)ϕ'(x)+1/2(Δx)2ϕ''(x)+..

Neglecting the higher than first order

  ϕnet

  (Δx)ϕ'(x) (1)

As the electric field is in the x direction

  ϕ(x)=ExΔyΔz

  ϕ'(x)=δExδxΔyΔz

Substituting in equation 1

  ϕnetΔxδExδxΔyΔz=δExδxΔxΔyΔz

  ϕnetδExδxΔV

Conclusion:

The net electric flux of the electric field out of the Gaussian surface is given by ϕnetδExδxΔV

b.

Show that δExδx=ρεo , where ρ is the volume charge density inside the cube.

  δExδx=ρεo

Given:

A small gaussian surface as shown

  Physics for Scientists and Engineers, Chapter 22, Problem 86P , additional homework tip  2

  ρ is the volume charge density inside the cube.

Formula Used:

Gauss’s law:

  ϕnet=qεo

Where, q is the charge enclosed

  εo is the permittivity of free space.

Calculations:

  ϕnet=qεo

  ϕnet=qεo=ρεoΔV

From part (a)

  ϕnetδExδxΔV

Equating the two equations

  δExδxΔV=ρεoΔV

  δExδx=ρεo

Conclusion:

  δExδx=ρεo

b.

To determine

Show that δExδx=ρεo , where ρ is the volume charge density inside the cube.

  δExδx=ρεo

Given:

A small gaussian surface as shown

  Physics for Scientists and Engineers, Chapter 22, Problem 86P , additional homework tip  3

  ρ is the volume charge density inside the cube.

Formula Used:

Gauss’s law:

  ϕnet=qεo

Where, q is the charge enclosed

  εo is the permittivity of free space.

Calculations:

  ϕnet=qεo

  ϕnet=qεo=ρεoΔV

From part (a)

  ϕnetδExδxΔV

Equating the two equations

  δExδxΔV=ρεoΔV

  δExδx=ρεo

Conclusion:

  δExδx=ρεo

Blurred answer
Students have asked these similar questions
what is the magnitude.of the electric flux of a constant E of 4 n/c in the z direction through a rectangle with surface area 4m^2 in the xy plane
An imaginary cubical surface of side LL has its edges parallel to the x-, y- and z-axes, one corner at the point x=0, y=0, z=0 and the opposite corner at the point x=L, y=L ,z=L. The cube is in a region of uniform electric field E⃗ =E1iˆ+E2jˆ, where E1E1 and E2E2 are positive constants. Calculate the electric flux through (a) the cube face in the plane z=0, (b) the cube face in the plane z=L, and (c) the entire cubical surface. For each face the normal points out of the cube.
Consider a cylindrical insulator of radius R and length L. This object has a surface charge density of σ(Φ) = a sin(5Φ) ( sigma(phi) = a sin(5(phi)) ) where a is a constant. If  L >> R, determine the electric field inside and outside the cylinder.

Chapter 22 Solutions

Physics for Scientists and Engineers

Knowledge Booster
Background pattern image
Physics
Learn more about
Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, physics and related others by exploring similar questions and additional content below.
Similar questions
SEE MORE QUESTIONS
Recommended textbooks for you
Text book image
Physics for Scientists and Engineers: Foundations...
Physics
ISBN:9781133939146
Author:Katz, Debora M.
Publisher:Cengage Learning
Text book image
Principles of Physics: A Calculus-Based Text
Physics
ISBN:9781133104261
Author:Raymond A. Serway, John W. Jewett
Publisher:Cengage Learning
Electric Fields: Crash Course Physics #26; Author: CrashCourse;https://www.youtube.com/watch?v=mdulzEfQXDE;License: Standard YouTube License, CC-BY