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Physics for Scientistsand Engineers

Chapter 22:The Electric Field II: Continuous

Charge Distributions

Copyright © 2004 by W. H. Freeman & Company

Paul A. Tipler • Gene Mosca

Fifth Edition

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Effect of Symmetry

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22-2Gauss’s Law

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Gauss’s Law

• Electric Flux

• Charge Distribution

• Relationship between field lines and charge

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Electric Flux

• E varies with density of lines

• Flux is #lines crossing a specific area

• Flux and “Flow”

• Symbol

• Units: N·m2/C

• Product of Field and Area

• Can be + or -

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Electric Flux (cont.)• Flux + when leaving a closed surface

• Flux - when entering a closed surface

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Electric Flux (cont.)• Notice that there is no charge inside

• and, Net Flux is zero

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Case where E is spatially uniform:

• = E·A (E factored out of integral)

• = +EA (E parallel to A)

• = -EA (E anti-parallel to A)

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Flux through both surfaces is identical

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Flux and Charge

• Amount and sign of a charge can be determined by

(#lines leaving) – (#lines entering)

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(#lines leaving) – (#lines entering) = 0net charge enclosed is zero

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net 8 lines leaving = net +q enclosed (with 8 lines per q)

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dAnnrEAdE ˆˆ)(

Flux due to a point Q

kQ

RR

kQdArE

4

)4()( 22

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•Net Flux not dependent on shape of enclosing surface or any charges outside the enclosure

•Net Flux does depend on amount of charge inside enclosure

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Cylindrical can enclosing part of an “infinite” plane of Q.

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Net flux = EA + EA + 0 = 2EA == 4kq

E = 4kq/2A = 2k(q/A) = 2k.

Plane of Charge cont.

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insidenet kQAdE 4

o

k4

1

Gauss’s Law

Permittivity of a vacuum,

o

Gauss’s Law

o

insideinsidenet

QkQ

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Gauss’s Law in terms of Permittivity

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Spherical Shell

• cosine = 1 (symmetry)

• = EA = Q/o

• E = Q/oA

• A = 4r2.

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any closed surface inside shell hasQenc = 0

EA ~ Q = 0

E = 0

Spherical Shell cont.

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“Field”: Concept or Reality?

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kQRLErightcylleftnet 4)2(

)(4)2(

)(4)2(

kRE

LkRLE

RRR

kE

oo

2

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4

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Long Line

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Uniform Spherical Volume

non-zero values inside

same as pt Q outside

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22-4Discontinuity of En

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kEo

n 41

o

k4

1

o

k

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22-5Charge and Field atConductor Surfaces

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E on Conductor

• at surface E = /o

• E normal (perpendicular) to surface

• E is zero inside (with static charges)

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+Point Q inside Shell

• shell = neutral conductor• -/+ induced on shell• E ~ same as for

lone +pt Q.

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Charge Distribution Field Shape

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Summary

• E obtained by sum of effect of all charges

• charges can be point (ch21) or ‘continuous’ (ch22)

• E can also be obtained by use of Gauss’s Law for E, where concept of E flux is used.

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22-6Derivation of Gauss’s Law

From Coulomb’s Law

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Problems

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