1 physics for scientists and engineers chapter 22: the electric field ii: continuous charge...
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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
12
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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