DepartmentofPhysicsandAppliedPhysicsPHYS.1440Lecture4A.Danylov

Lecture 4

Chapter 24

Electric FluxGauss’s Law is like Yankee swap

Finally! Some fun!

Physics II

Course website:https://sites.uml.edu/andriy-danylov/teaching/physics-ii/

DepartmentofPhysicsandAppliedPhysicsPHYS.1440Lecture4A.Danylov

Today we are going to discuss:

Chapter 24:

Section24.2:Idea of Flux Section24.3:Electric Flux Section24.4:Gauss’s Law

DepartmentofPhysicsandAppliedPhysicsPHYS.1440Lecture4A.Danylov

Gauss’s Law

DepartmentofPhysicsandAppliedPhysicsPHYS.1440Lecture4A.Danylov

The idea behind Gauss’s law

Itlookslikenumberoflinespassingthroughaclosedsurfacearerelatedtotheamountofchargeinside(Gauss’sLaw)

But,first,weneedtolearnhowtocountlines Flux

DepartmentofPhysicsandAppliedPhysicsPHYS.1440Lecture4A.Danylov

The Basic Idea of Flux

Thevolumeofairflowingthroughtheloopeachseconddependsontheanglebetweentheloopandthedirectionofflow.

Theflowismaximumthroughaloopthatisperpendiculartotheairflow.

Noairgoesthroughthesameloopifitliesparalleltotheflow.

ImagineholdingarectangularwireloopofareaAinfrontofafan.

DepartmentofPhysicsandAppliedPhysicsPHYS.1440Lecture4A.Danylov

The Area Vector

Let’s define an area vector to be a vector in the direction of , perpendicular to the surface, with a magnitude A equal to the area of the surface.

Vector has units of m2.

Before defining the electric flux, we need to introduce the area vector.

DepartmentofPhysicsandAppliedPhysicsPHYS.1440Lecture4A.Danylov

The Electric Flux

The electric flux through a surface of area A can be defined as the dot-product:

θ

Consider 1) a uniform electric field E 2) a flat surface

TheelectricfluxthroughtheshadedsurfaceisConcepTest Electric Flux

A) 0

B) 200 N m2/C

C) 400 N m2/C

D) Some other value

TheelectricfluxthroughtheshadedsurfaceisConcepTest Electric Flux

A) 0

B)200Nm2/C

C)400Nm2/C

D)Someothervalue

Theelectricfluxthroughtheshadedsurfaceis

ConcepTest Electric FluxA) 0B)400cos20 Nm2/CC)400cos70 Nm2/CD)400Nm2/CE)Someothervalue

DepartmentofPhysicsandAppliedPhysicsPHYS.1440Lecture4A.Danylov

The Electric Flux (general case)

Φ ∙

Consider1)anon‐uniformelectricfieldE

2)areaisnotflat(curved)

DividethesurfaceintomanysmallpiecesofareaA.

Theelectricfluxthrougheachsmallpieceis:

Theelectricfluxthroughthewholesurfaceisthesurfaceintegral:

a)Eachpieceissmallenoughthatitisessentiallyflatb)Thefieldisnearlyuniformovereachpiecec)Thus,theformulafromthepreviousslidecanbeused

DepartmentofPhysicsandAppliedPhysicsPHYS.1440Lecture4A.Danylov

The Electric Flux through a closed surface

Consider1)anon‐uniformelectricfieldE2)aclosedsurface

Theelectricfluxthroughaclosedsurface:

NOTE:Foraclosedsurface,weusetheconventionthattheareavectordA isdefinedtoalwayspointtowardtheoutside.

closed surface

DepartmentofPhysicsandAppliedPhysicsPHYS.1440Lecture4A.Danylov

Φ ∙

Gauss’s Law

ForanyclosedsurfaceenclosingtotalchargeQin,thenetelectricfluxthroughthesurfaceis:

This result for the electric flux is known as Gauss’s Law.

q1

q3

q2

Gaussian surface

q5

q4

1)ItworksforanyclosedsurfacecalledGaussian

2)QinisthenetchargeenclosedbytheGaussiansurface(chargesoutsidemustnotbeincluded)

3)DistributionofQindoesn'tmatter

(AGaussiansurfaceisanimaginary,mathematicalsurface)

Gaussian surface

Both, Gauss’s law and Coulomb’s law, help to find electric fields based on distribution of charges.

Properties:

dA

E

DepartmentofPhysicsandAppliedPhysicsPHYS.1440Lecture4A.Danylov

Gauss’s Law/Symmetry

Evaluationofthissurfaceintegralisoftendifficult.However,whenthechargedistributionhassufficientsymmetry(spherical,cylindrical,planar),evaluationof

theintegralbecomessimple.

Φ ∙Gaussian surface

DepartmentofPhysicsandAppliedPhysicsPHYS.1440Lecture4A.Danylov

Thank you