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

qq

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