Chapter 28. Gauss's Law

Copyright © 2008 Pearson Education, Inc., publishing as Pearson Addison-Wesley.

Chapter 28. Gauss’s Law

The nearly spherical shape

of the girl’s head determines

the electric field that causes

her hair to stream outward.

Using Guass’s law, we can

deduce electric fields,

particularly those with a

high degree of symmetry,

simply from the shape of the

charge distribution.

Chapter Goal: To

understand and apply

Gauss’s law.


Topics:

• The Concept of Flux

• Gauss’s Law

• Conductors in Electrostatic Equilibrium

• Calculating Electric Flux

• Symmetry

• Using Gauss’s Law

Chapter 28. Gauss’s Law


Chapter 28. Reading Quizzes


The amount of electric field

passing through a surface is

called

A. Electric flux.

B. Gauss’s Law.

C. Electricity.

D. Charge surface density.

E. None of the above.


The amount of electric field

passing through a surface is

called

A. Electric flux.

B. Gauss’s Law.

C. Electricity.

D. Charge surface density.

E. None of the above.


Gauss’s law is useful for calculating

electric fields that are

A. symmetric.

B. uniform.

C. due to point charges.

D. due to continuous charges.


Gauss’s law is useful for calculating

electric fields that are

A. symmetric.

B. uniform.

C. due to point charges.

D. due to continuous charges.


Gauss’s law applies to

A. lines.

B. flat surfaces.

C. spheres only.

D. closed surfaces.


Gauss’s law applies to

A. lines.

B. flat surfaces.

C. spheres only.

D. closed surfaces.


The electric field inside a

conductor in electrostatic

equilibrium is

A. uniform.

B. zero.

C. radial.

D. symmetric.


The electric field inside a

conductor in electrostatic

equilibrium is

A. uniform.

B. zero.

C. radial.

D. symmetric.


Chapter 28. Basic Content and Examples


The Electric Flux

The electric flux measures the amount of electric field

passing through a surface of area A whose normal to the

surface is tilted at angle from the field.

We can define the electric flux more concisely using the

dot-product:



EXAMPLE 28.1 The electric flux inside a

parallel-plate capacitor

QUESTION:









Tactics: Evaluating surface integrals


The Electric Flux through a Closed Surface

The electric flux through a closed surface is

The electric flux is still the summation of the fluxes through

a vast number of tiny pieces, pieces that now cover a closed

surface.

NOTE: A closed surface has a distinct inside and outside.

The area vector dA is defined to always point toward the

outside. This removes an ambiguity that was present for a

single surface, where dA could point to either side.


Gauss’s Law

For any closed surface enclosing total charge Qin,the net

electric flux through the surface is

This result for the electric flux is known as Gauss’s Law.


Electrostatics of Conductors

Using the Gauss’ law, we can derive a set of important

properties about conductors in electrostatic

equilibrium.

(You may claim that you understand a little electrostatics

if you know all the following results. )


Interior of Conductors in Electrostatic

Equilibrium

1. The electric field is zero at all points within aconductor in electrostatic equilibrium.Otherwise, the electric field would cause electrons to move and thus

violate the assumption that all the charges are at rest.

2. There can be no net charge inside any conductor inelectrostatic equilibrium.Otherwise, we construct a Gaussian surface around the region with

nonvanishing net charge, and we must have non-zero field on the

surface--in contradiction with result 1.

---> The inside of a conductor is completely “empty” asfar as electrostatics is concerned.


Boundary of Conductors in Electrostatic

Equilibrium

3. All charges on a conductor must be distributed on its

surface: “surface charge”.

The electric field outside of conductor can be nonzero.

4. Right outside a conductor, the tangent component of

the field must be vanish: otherwise surface charge will

move.

--> The field on the conductor surface

must be perpendicular to the surface.


The Normal Component of Field

at the outer-surface of Conductor

a). Construct a cylindrical Gaussian

surface and integrate the field over.

b). The flux through the inner disk and

side surface is clearly zero

c). The flux through the outer disk is

given by E A.

d). The total flux through the cylinder

Is given by the Gauss’ law.

5. The magnitude of electric field on the outer-surface of a

conductor is proportional to the surface charge density.


Cavity inside a Conductor

In the presence of a cavity, again there can be no net charge

in the interior of a conductor.

Documents

Chapter 28. Gauss's Law