Post on 17-Jan-2016
transcript
Gauss’ Law
Electric Field Lines / Electric Field VectorsElectric FluxGauss’ LawUse of Gauss’ Law and Gaussian SurfacesElectrostatic Equilibrium
#Conductors
#Non Conductors
Electric Field Vectors and Lines
Electric Force and AccelerationThe electric force is
given by F = qEThe acceleration by
a qm
E
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
EA E A NC
2
m NC
2
m
Definition
Flux Picture
Flux Picture
Area VectorDefine Area Vector
A An
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
Dot product Definition of Flux
A E
AECos
Electric FluxNumber of Field lines through Perpendicular surface
Flux through closed surfaceFlux through a
closedclosed
surface from ansurface from an
external sourceexternal source
is zerois zero
Closed Surface Picture
Surface Area Element
Flux through Curved Surface E dA
surface
E dAEdA Cos
A dAsurface
Spherical Surface
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
Flux through any closed surface surrounding a
charge is the same
Gauss' Law I E d AGaussian surface
E r dAGaussian surface
E r dA
Gaussian surface
E r 4 r 2
Gauss' Law III
k Qr 2
4 r 2
4 kQ Q
0
Using Coulombs Law for a point charge
Gauss' Law IIGauss’ Law
E dAGaussian surface
Q
0
To Find Electric Field of Given Charge Distribution
Surface + Charge
Field
Use of Gauss' Law
Closed Surfaces
Coulombs Law from Gauss' Law IGauss' Law
Coulombs' Law
Coulombs Law from Gauss' Law I
2
02
2
radiusof sphere
radiusof sphere0
4
4
r
Qk
r
QrE
rrEdArE
drQ
r
r
AE
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
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
# is perpendicular to conductors surface# has magnitude =surface density / permitivity
Electric Field just outside conductor
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)
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?
AnswerCharge must be
between surface 1 and surface 2
(why?)Therefore must be on the
surface of object
What is Electric Field on surface?
1
23
•Zero Flux through 2•Zero Flux through 3•Only Flux through 1
E
Answer
Answer 2Qinside
cylinder
0
E dAcylinder
E r dA
disk 1
E r A
E r Qinside
cylinder
A 0
r
0
Answer 3Direction of Field?
Must be orthogonal to surface
otherwise there will be net motion on surface
magnitude of electric
field
distance from center
of charged conductor
radius of conductor
Graph of Field v. Position
In external field conductor becomes polarizedpolarized InducedInduced Electric Field
from the surface must cancel external Electric Field inside conductor
Conductor in Electric Field
Induced Field
E
E
E
Eq
q
q
q
Einduced
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
Electric Field inside Cavities
Electric Fields inside Cavities of Cavities of Conductors Conductors Gaussian Gaussian
SurfaceSurface
CavityCavity
Analysis 1
Total charge within Gaussian surface must be
zeroOtherwise there is an
Electric Field inside the conductor around the cavity
Therefore NO charge on surface of cavity
Can enlarge cavity so that conductor is hollow
Faraday cageFaraday cage
Analysis 2
Radio receptionover some
bridges
Thought Question
Electric Field inside NonconductorElectric
Field inside non
conductor?
magnitude of electric field
distance from center of charged non conductor
radius of non conductor
Graph of Field v. Position
Field Above ConductorField above surface of
charged conductor
Does not depend on thickness of conductor
E QA 0
0
Field Above Very Thin Nonconductor
Field above surface of charged nonconductor
00 22
2
A
QE
EA