gauss’ law áelectric field lines / electric field vectors áelectric flux ágauss’ law áuse of...
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Gauss’ Law
Electric Field Lines / Electric Field VectorsElectric FluxGauss’ LawUse of Gauss’ Law and Gaussian SurfacesElectrostatic Equilibrium
#Conductors
#Non Conductors
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Electric Field Vectors and Lines
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Electric Force and AccelerationThe electric force is
given by F = qEThe acceleration by
a qm
E
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A measure of the amount of electric field through an area perpendicular to the fieldThe “number” of field lines through the area.
Electric FluxElectric Flux
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EA E A NC
2
m NC
2
m
Definition
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Flux Picture
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Flux Picture
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Area VectorDefine Area Vector
A An
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Definition of symbols
A = Area (always positive number)n = Unit vector. Its direction corresponds to the orientation of the area Forms a right handed system
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Dot product Definition of Flux
A E
AECos
Electric FluxNumber of Field lines through Perpendicular surface
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Flux through closed surfaceFlux through a
closedclosed
surface from ansurface from an
external sourceexternal source
is zerois zero
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Closed Surface Picture
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Surface Area Element
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Flux through Curved Surface E dA
surface
E dAEdA Cos
A dAsurface
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Spherical Surface
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Gaussian SurfaceGaussian Surface defined as
Surface
# surrounding surrounding charge
# where magnitude magnitude of Electric Field is constant constant
or zero
# the directiondirection of Electric Field is
same as the Area vectors Area vectors of the surface
# thus same symmetrysymmetry as charge
distribution
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Flux through any closed surface surrounding a
charge is the same
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Gauss' Law I E d AGaussian surface
E r dAGaussian surface
E r dA
Gaussian surface
E r 4 r 2
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Gauss' Law III
k Qr 2
4 r 2
4 kQ Q
0
Using Coulombs Law for a point charge
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Gauss' Law IIGauss’ Law
E dAGaussian surface
Q
0
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To Find Electric Field of Given Charge Distribution
Surface + Charge
Field
Use of Gauss' Law
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Closed Surfaces
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Coulombs Law from Gauss' Law IGauss' Law
Coulombs' Law
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Coulombs Law from Gauss' Law I
2
02
2
radiusof sphere
radiusof sphere0
4
4
r
Qk
r
QrE
rrEdArE
drQ
r
r
AE
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Electrostatic EquilibriumElectrostatic Equilibrium
for objects in an external Electric Field
Conductors# No net motion of charge within conductor
Non Conductors# in non conductors there is no movement of charge# therefore always have equilibrium
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At Electrostatic Equilibrium
At Electrostatic Equilibrium
Electric Field is zero within conductor
Any excess charge on an isolated conductor must be on its surface
# accumulates at points where radius of curvature is greatest
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# is perpendicular to conductors surface# has magnitude =surface density / permitivity
Electric Field just outside conductor
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Electric Field inside conductor Net Electric Field is zero
inside, otherwise Net Electric
Force on charges which then accelerate
and move charges (on the average)
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Why is the Charge on the Surface?
QE=0
Gaussian Surface 1
Gaussian Surface 2
Use Gauss’ Theorem
Why is the charge on the surface?
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AnswerCharge must be
between surface 1 and surface 2
(why?)Therefore must be on the
surface of object
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What is Electric Field on surface?
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1
23
•Zero Flux through 2•Zero Flux through 3•Only Flux through 1
E
Answer
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Answer 2Qinside
cylinder
0
E dAcylinder
E r dA
disk 1
E r A
E r Qinside
cylinder
A 0
r
0
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Answer 3Direction of Field?
Must be orthogonal to surface
otherwise there will be net motion on surface
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magnitude of electric
field
distance from center
of charged conductor
radius of conductor
Graph of Field v. Position
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In external field conductor becomes polarizedpolarized InducedInduced Electric Field
from the surface must cancel external Electric Field inside conductor
Conductor in Electric Field
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Induced Field
E
E
E
Eq
q
q
q
Einduced
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If the conductor has a net charge
then it is also a source of an Electric Field
that combines with the external field
producing a resultant field
external to the conductor
Charged Conductor
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Electric Field inside Cavities
Electric Fields inside Cavities of Cavities of Conductors Conductors Gaussian Gaussian
SurfaceSurface
CavityCavity
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Analysis 1
Total charge within Gaussian surface must be
zeroOtherwise there is an
Electric Field inside the conductor around the cavity
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Therefore NO charge on surface of cavity
Can enlarge cavity so that conductor is hollow
Faraday cageFaraday cage
Analysis 2
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Radio receptionover some
bridges
Thought Question
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Electric Field inside NonconductorElectric
Field inside non
conductor?
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magnitude of electric field
distance from center of charged non conductor
radius of non conductor
Graph of Field v. Position
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Field Above ConductorField above surface of
charged conductor
Does not depend on thickness of conductor
E QA 0
0
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Field Above Very Thin Nonconductor
Field above surface of charged nonconductor
00 22
2
A
QE
EA