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Understanding Electric Flux and Gauss' Law in Physics

This lecture covers electric charge, the electric field, moving charges, electronic circuit components, magnetic field, time-varying magnetic field, electromagnetic waves, and optics.

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Understanding Electric Flux and Gauss' Law in Physics

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  1. Physics 2102 Jonathan Dowling Physics 2102 Lecture: 05 FRI 23 JAN Flux Capacitor (Schematic) Gauss’ Law I Carl Friedrich Gauss 1777 – 1855 Version: 1/22/07

  2. What Are We Going to Learn?A 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. • Electromagneticwaves-light waves • Geometrical Optics (light rays). • Physical optics (light waves)

  3. STRONG E-Field Angle Matters Too Weak E-Field  dA Number of E-Lines Through Differential Area “dA” is a Measure of Strength What? — The Flux!

  4. E q normal AREA = A=An Electric Flux: Planar Surface • Given: • planar surface, area A • uniform field E • E makes angle q with NORMAL to plane • Electric Flux: F = E•A = E A cosq • Units: Nm2/C • Visualize: “Flow of Wind” Through “Window”

  5. E dA E Area = dA dA Electric Flux: General Surface • For any general surface: break up into infinitesimal planar patches • Electric FluxF = EdA • Surface integral • dAis a vector normal to each patch and has a magnitude = |dA|=dA • CLOSED surfaces: • define the vector dA as pointing OUTWARDS • Inward E gives negative fluxF • Outward E gives positive fluxF

  6. dA (pR2)E–(pR2)E=0 What goes in — MUST come out! dA Electric Flux: Example E • Closed cylinder of length L, radius R • Uniform E parallel to cylinder axis • What is the total electric flux through surface of cylinder? (a) (2pRL)E (b) 2(pR2)E (c) Zero L R Hint! Surface area of sides of cylinder: 2pRL Surface area of top and bottom caps (each): pR2

  7. dA 1 2 dA 3 dA Electric Flux: Example • Note that E is NORMAL to both bottom and top cap • E is PARALLEL to curved surface everywhere • So: F = F1+ F2 + F3 =pR2E + 0 – pR2E = 0! • Physical interpretation: total “inflow” = total “outflow”!

  8. Electric Flux: Example • Spherical surface of radius R=1m; E is RADIALLY INWARDS and has EQUAL magnitude of 10 N/C everywhere on surface • What is the flux through the spherical surface? • (4/3)pR2 E = -13.33p Nm2/C (b) 2pR2 E = -20p Nm2/C (c) 4pR2 E= -40p Nm2/C What could produce such a field? What is the flux if the sphere is not centered on the charge?

  9. r q Electric Flux: Example (Inward!) (Outward!) Since r is Constant on the Sphere — Remove E Outside the Integral! Surface Area Sphere Gauss’ Law: Special Case!

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