Class 05: OutlineSix PRS Questions On Pace and PreparationLast Time:Potential and E FieldE Field and Potential: CreatingE Field and Potential: EffectsTwo PRS Questions:Potential & E FieldGauss’s LawGauss’s Law – The IdeaGauss’s Law – The EquationNow the DetailsElectric Flux FEElectric Flux FEPRS Question:Flux Thru SheetGauss’s LawOpen and Closed SurfacesArea Element dA: Closed SurfaceElectric Flux FEExample: Point ChargeOpen SurfaceExample: Point ChargeClosed SurfacePRS Question:Flux Thru SphereElectric Flux: SphereArbitrary Gaussian SurfacesApplying Gauss’s LawChoosing Gaussian SurfaceSymmetry & Gaussian SurfacesPRS Question:Should we use Gauss’ Law?Gauss: Spherical SymmetryGauss: Spherical SymmetryGauss: Spherical SymmetryGauss: Spherical SymmetryGauss: Spherical SymmetryPRS Question:Field Inside Spherical ShellGauss: Cylindrical SymmetryGauss: Cylindrical SymmetryGauss: Cylindrical SymmetryGauss: Planar SymmetryGauss: Planar SymmetryGauss: Planar SymmetryPRS Question:Slab of ChargeGroup Problem: Charge SlabPRS Question:Slab of ChargePotential from EPotential for Uniformly Charged Non-Conducting Solid SpherePotential for Uniformly Charged Non-Conducting Solid SpherePotential for Uniformly Charged Non-Conducting Solid SphereGroup Problem: Charge SlabGroup Problem: Spherical Shells1P05 -Class 05: OutlineHour 1:Gauss’ LawHour 2:Gauss’ Law2P05 -Six PRS Questions On Pace and Preparation3P05 -Last Time:Potential and E Field4P05 -E Field and Potential: CreatingA point charge q creates a field and potential around it:2ˆ;eeqqkVkrr==ErGUse superposition for systems of charges;BBAAVVVV d=−∇ ∆ ≡ − =− ⋅∫EEsGGGThey are related:5P05 -E Field and Potential: Effectsq=FEGGIf you put a charged particle, q, in a field:To move a charged particle, q, in a field:WUqV=∆=∆6P05 -Two PRS Questions:Potential & E Field7P05 -Gauss’s LawThe first Maxwell Equation A very useful computational technique This is important!8P05 -Gauss’s Law – The IdeaThe total “flux” of field lines penetrating any of these surfaces is the same and depends only on the amount of charge inside9P05 -Gauss’s Law – The Equation0SsurfaceclosedεinEqd =⋅=Φ∫∫AEGGElectric flux ΦE(the surface integral of E over closed surface S) is proportional to charge inside the volume enclosed by S10P05 -Now the Details11P05 -Electric Flux ΦECase I: E is constant vector field perpendicular to planar surface S of area A∫∫⋅=Φ AEGGdEEEAΦ=+Our Goal: Always reduce problem to this12P05 -Electric Flux ΦECase II: E is constant vector field directed at angle θto planar surface S of area AcosEEAθΦ=∫∫⋅=Φ AEGGdE13P05 -PRS Question:Flux Thru Sheet14P05 -Gauss’s Law0SsurfaceclosedεinEqd =⋅=Φ∫∫AEGGNote: Integral must be over closed surface15P05 -Open and Closed SurfacesA rectangle is an open surface — it does NOT contain a volume A sphere is a closed surface — it DOES contain a volume16P05 -Area Element dA: Closed SurfaceFor closed surface, dA is normal to surface and points outward( from inside to outside)ΦE > 0 if E points outΦE < 0 if E points in17P05 -Electric Flux ΦECase III: E not constant, surface curved EddΦ= ⋅EAGGSAdE∫∫Φ=ΦEEd18P05 -Example: Point ChargeOpen Surface19P05 -Example: Point ChargeClosed Surface20P05 -PRS Question:Flux Thru Sphere21P05 -Electric Flux: Sphere Point charge Q at center of sphere, radius rE field at surface:20ˆ4Qrπε=ErGElectric flux through sphere:20Sˆˆ4QdArπε=⋅∫∫rrwSEdΦ= ⋅∫∫EAGGwrAˆdAd =G0Qε=22044Qrrππε=20S4QdArπε=∫∫w22P05 -Arbitrary Gaussian Surfaces 0SsurfaceclosedεQdE=⋅=Φ∫∫AEGGFor all surfaces such as S1, S2or S323P05 -Applying Gauss’s Law1. Identify regions in which to calculate E field.2. Choose Gaussian surfaces S: Symmetry3. Calculate 4. Calculate qin, charge enclosed by surface S5. Apply Gauss’s Law to calculate E:0SsurfaceclosedεinEqd =⋅=Φ∫∫AEGG∫∫⋅=ΦSAEGGdE24P05 -Choosing Gaussian SurfaceChoose surfaces where E is perpendicular & constant. Then flux is EA or -EA.Choose surfaces where E is parallel.Then flux is zeroOREAEA−EExample: Uniform FieldFlux is EA on topFlux is –EA on bottomFlux is zero on sides25P05 -Symmetry & Gaussian SurfacesSymmetry Gaussian SurfaceSpherical Concentric SphereCylindrical Coaxial CylinderPlanar Gaussian “Pillbox”Use Gauss’s Law to calculate E field from highly symmetric sources26P05 -PRS Question:Should we use Gauss’ Law?27P05 -Gauss: Spherical Symmetry+Q uniformly distributed throughout non-conducting solid sphere of radius a. Find E everywhere28P05 -Gauss: Spherical SymmetrySymmetry is Spherical Use Gaussian SpheresrEˆE=G29P05 -Gauss: Spherical SymmetryRegion 1: r > aDraw Gaussian Sphere in Region 1 (r > a)Note: r is arbitrary but is the radius for which you will calculate the E field!30P05 -Gauss: Spherical SymmetryRegion 1: r > aTotal charge enclosed qin= +QSEdA EA==∫∫wSEdΦ= ⋅∫∫EAGGw()24Erπ=2004inEqQrEπεεΦ= = =204QErπε=20ˆ4Qrπε⇒=ErG31P05 -Gauss: Spherical SymmetryGauss’s law:QarQarqin⎟⎟⎠⎞⎜⎜⎝⎛=⎟⎟⎟⎟⎠⎞⎜⎜⎜⎜⎝⎛=33333434ππRegion 2: r < aTotal charge enclosed:()24EErπΦ=304QrEaπε=30ˆ4Qraπε⇒=ErGORinqVρ=3300inqrQaεε⎛⎞==⎜⎟⎝⎠32P05 -PRS Question:Field Inside Spherical Shell33P05 -Gauss: Cylindrical SymmetryInfinitely long rod with uniform charge density λFind E outside the rod.34P05 -Gauss: Cylindrical SymmetrySymmetry is Cylindrical Use Gaussian CylinderrEˆE=GNote: r is arbitrary but is the radius for which you will calculate the E field!A is arbitrary and should divide out35P05 -Gauss: Cylindrical SymmetryAλ=inqTotal charge enclosed:()002inqErλπεε===AASSEdEdAEAΦ= ⋅ = =∫∫ ∫∫EAGGww02Erλπε=0ˆ2rλπε⇒=ErG36P05 -Gauss: Planar SymmetryInfinite slab with uniform charge density σFind E outside the plane37P05 -Gauss: Planar SymmetrySymmetry is Planar Use Gaussian PillboxxEˆE±=GxˆGaussianPillboxNote: A is arbitrary (its size and shape) and should divide out38P05 -Gauss: Planar SymmetryAqinσ=Total charge enclosed:NOTE: No flux through side of cylinder, only endcaps()002inqAEAσεε===SSE EndcapsdEdAEAΦ= ⋅ = =∫∫ ∫∫EAGGww++++++++++++σEGEGxA{}0ˆto rightˆ-to left2σε⇒=xExG02Eσε=39P05 -PRS Question:Slab of Charge40P05 -Group Problem: Charge SlabInfinite slab with uniform charge density ρThickness is 2d (from x=-d to x=d). Find E everywhere.xˆ41P05 -PRS Question:Slab of
View Full Document
Unlocking...