DepartmentofPhysicsandAppliedPhysicsPHYS.1440Lecture5 A.Danylov

Lecture 5

Chapter 24

Gauss’s Law

95.144

Physics II

What? Surface integration?He must be kidding!

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

DepartmentofPhysicsandAppliedPhysicsPHYS.1440Lecture5 A.Danylov

Today we are going to discuss:Chapter 24:

Section24.4:Gauss’s Law

DepartmentofPhysicsandAppliedPhysicsPHYS.1440Lecture5 A.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.1440Lecture5 A.Danylov

Gauss’s Law/Symmetry

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

theintegralbecomessimple.

Φ ∙Gaussian surface

DepartmentofPhysicsandAppliedPhysicsPHYS.1440Lecture5 A.Danylov

Notice a remarkable feature of this result:The field outside the sphere isexactly the same as it would have beenif all the charge had beenconcentrated at the center.

A total charge Q is spread uniformly throughout a dielectric sphere of radius R. What is the electric field outside the sphere (r>R)?

Example E field outside of a charged dielectric sphere

Example 24.3

DepartmentofPhysicsandAppliedPhysicsPHYS.1440Lecture5 A.Danylov

A total charge Q is spread uniformlythroughout a dielectric sphere ofradius R. What is the electric fieldinside the sphere (r<R)?

ELet’s plot it

Example E field inside of a charged dielectric sphere

Example 24.4

DepartmentofPhysicsandAppliedPhysicsPHYS.1440Lecture5 A.Danylov

Suppose the nonconducting sphere hasa spherical cavity of radius a centeredat the sphere’s center. Assuming thecharge Q is distributed uniformly inthe “shell” (between r = b and r = a),determine the electric field as afunction of r fora) r < ab) a < r < bc) b < r

Example E field created by a sphere with a spherical cavity

DepartmentofPhysicsandAppliedPhysicsPHYS.1440Lecture5 A.Danylov

Findtheelectricfieldofaninfinitelylonglineofchargewithlinearcharge

densityλ (C/m).

Example 24.4

Example E field outside of a long, charged wire

DepartmentofPhysicsandAppliedPhysicsPHYS.1440Lecture5 A.Danylov

Find the electric field of an infinitely long line of charge with volume charge density ρ (C/m).

Problem 24.50(b)

Example E field inside of a long, charged wire

DepartmentofPhysicsandAppliedPhysicsPHYS.1440Lecture5 A.Danylov

Thank you