Physics 2102Physics 2102Lecture: 05 FRI 23 JANLecture: 05 FRI 23 JANGaussGauss’’ Law ILaw IVersion: 1/22/07Flux Capacitor (Schematic)Physics 2102Jonathan DowlingCarl Friedrich Gauss1777 – 1855What Are We Going to Learn?What Are We Going to Learn?A Road MapA Road Map• Electric charge- Electric force on other electric charges- Electric field, and electric potential• Moving electric charges : current• Electronic circuit components: batteries, resistors,capacitors• Electric currents - Magnetic field- Magnetic force on moving charges• Time-varying magnetic field - Electric Field• More circuit components: inductors.• Electromagnetic waves - light waves• Geometrical Optics (light rays).• Physical optics (light waves)What? What? —— The Flux! The Flux!STRONGE-FieldWeakE-FieldNumber of E-LinesThrough DifferentialArea “dA” is a Measureof StrengthdAqAngleMatters TooElectric Flux: Planar SurfaceElectric Flux: Planar Surface• Given:– planar surface, area A– uniform field E– E makes angle q with NORMAL toplane• Electric Flux:Φ = E•A = E A cosθ• Units: Nm2/C• Visualize: “Flow of Wind”Through “Window”θEAREA = A=AnnormalElectric Flux: General SurfaceElectric Flux: General Surface• For any general surface: break upinto infinitesimal planar patches• Electric Flux Φ = ∫E•dA• Surface integral• dA is a vector normal to eachpatch and has a magnitude =|dA|=dA• CLOSED surfaces:– define the vector dA as pointingOUTWARDS– Inward E gives negative flux Φ – Outward E gives positive flux ΦEdAdAEArea = dAElectric Flux: ExampleElectric Flux: Example• Closed cylinder of length L, radius R• Uniform E parallel to cylinder axis• What is the total electric flux throughsurface of cylinder?(a) (2πRL)E(b) 2(πR2)E(c) ZeroHint!Surface area of sides of cylinder: 2π RLSurface area of top and bottom caps (each): πR2LRE(πR2)E–(πR2)E=0What goes in — MUST come out! dAdAElectric Flux: ExampleElectric Flux: Example• Note that E isNORMAL to bothbottom and top cap• E is PARALLEL tocurved surfaceeverywhere• So: Φ = Φ1+ Φ2 + Φ3 = πR2E + 0 – πR2E= 0!• Physical interpretation:total “inflow” = total“outflow”!3dA1dA2dAElectric Flux: ExampleElectric Flux: Example• Spherical surface of radius R=1m; E is RADIALLYINWARDS and has EQUAL magnitude of 10 N/Ceverywhere on surface• What is the flux through the spherical surface?(a) (4/3)πR2 E = −13.33π Nm2/C(b) 2πR2 E = −20π Nm2/C(c) 4π R2 E= −40π Nm2/CWhat could produce such a field?What is the flux if the sphere is notcentered on the charge?qrElectric Flux: ExampleElectric Flux: Example ! d! A = + dA( )ˆ r ! ! E • d! A = EdA cos(180°) = "EdA ! ! E = "qr2ˆ r Since r is Constant on the Sphere — RemoveE Outside the Integral! ! " =! E # d! A $= %E dA = %kqr2& ' ( ) * + $4,r2( )= %q4,-04,( )= %q /-0Gauss’ Law:Special Case!(Outward!)Surface Area
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