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

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