PHY2049: Chapter 231Chapter 23: Gauss’ LawPHY2049: Chapter 232Electric FluxÎSimple definition of electric flux (E constant, flat surface) E at an angle θ to planar surface, area A Units = N m2/ C (SI units)ÎSimple example Let E = 104N/C pass through 2m x 5m rectangle, 30° to normal φE= 104* 10 * cos(30) = 100,000 * 0.866 = 86,600ÎMore general ΦEdefinition (E variable, curved surface)cosEEAθΦ≡⋅ =EAESdΦ≡ ⋅∫EANormalEPHY2049: Chapter 233Example of Constant Field()2ˆˆ23mAij=+ˆ4Ei=()ˆˆˆ2348EijiΦ≡⋅ = + ⋅ =EAPHY2049: Chapter 234Flux Through Closed SurfaceÎSurface elements dAalways point outward!ÎSign of ΦEE outward (+) E inward (−)ΦE< 0ΦE= 0ΦE> 0PHY2049: Chapter 235Example: Flux Through CubeÎE field is constant: Flux through front face? Flux through back face? Flux through top face? Flux through whole cube?ˆEEz=GPHY2049: Chapter 236Example: Flux Through CylinderÎAssume E is constant, to the right Flux through left face? Flux through right face? Flux through curved side Total flux through cylinder?PHY2049: Chapter 237Example: Flux Through SphereÎAssume point charge +QÎE points radially outward (normal to surface!)()22044ESdkQrrQkQππεΦ= ⋅⎛⎞=⎜⎟⎝⎠==∫EAForeshadowing of Gauss’ Law!PHY2049: Chapter 238Gauss’ LawÎGeneral statement of Gauss’ lawÎCan be used to calculate E fields. But remember Outward E field, flux > 0 Inward E field, flux < 0ÎConsequences of Gauss’ law (as we shall see) E = 0 inside conductor E is always normal to surface on conductor Excess charge on conductor is always on surfaceConductorenc0Sqdε⋅=∫EAvIntegration over closedsurfaceqencis charge insidethe surfaceCharges outside surface have no effectPHY2049: Chapter 239Reading QuizÎWhat is the electric flux through a sphere of radius R surrounding a charge +Q at the center? 1) 0 2) +Q/ε0 3) −Q/ε0 4) +Q 5) None of thesePHY2049: Chapter 2310QuestionHow does the flux ФEthrough the entire surface change when the charge +Q is moved from position 1 to position 2?a) ФE increasesb) ФE decreasesc) ФE doesn’t changedSdS12+Q+QJust depends on chargenot positionPHY2049: Chapter 2311Power of Gauss’ Law: Calculating E FieldsÎValuable for cases with high symmetry E = constant, ⊥ surface E || surfaceÎSpherical symmetry E field vs r for point charge E field vs r inside uniformly charged sphere Charges on concentric spherical conducting shellsÎCylindrical symmetry E field vs r for line charge E field vs r inside uniformly charged cylinderÎRectangular symmetry E field for charged plane E field between conductors, e.g. capacitorsSdEA⋅=∫EAv0Sd⋅=∫EAvPHY2049: Chapter 2312ExampleÎ4 Gaussian surfaces: 2 cubes and 2 spheresÎRank magnitudes of E field on surfacesÎWhich ones have variable E fields?ÎWhat are the fluxes over each of the Gaussian surfaces(a) E falls as radius increases(b) E non-constant on cube (r changes)(c) Fluxes are same, +Q/ε0PHY2049: Chapter 2313Derive Coulomb’s Law From Gauss’ LawÎCharge +Q at a point By symmetry, E must be radially symmetricÎDraw Gaussian’ surface around point Sphere of radius r E field has constant mag., ⊥ to Gaussian surfaceGaussian surface(sphere)r()204SQdErπε⋅= =∫EAv2204QkQErrπε==Gauss’ LawSolve for EPHY2049: Chapter 2314ExampleÎCharges on shells are +Q (ball at center) +3Q (middle shell) +5Q (outside shell)ÎFind fluxes on the three Gaussian surfaces(a) Inner +Q/ε0(b) Middle +4Q/ε0(c) Outer
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