Physics 2102 Jonathan Dowling Flux Capacitor Schematic Physics 2102 Lecture 3 Gauss Law I Michael Faraday 1791 1867 Version 1 22 07 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 Electromagnetic waves light waves Geometrical Optics light rays Physical optics light waves What The Flux STRONG E Field Angle Matters Too Weak E Field dA Number of E Lines Through Differential Area dA is a Measure of Strength Electric Flux Planar Surface Given planar surface area A uniform field E E makes angle with NORMAL to plane Electric Flux E A E A cos Units Nm2 C Visualize Flow of Wind Through Window E normal AREA A An Electric Flux General Surface For any general surface break up into infinitesimal planar patches Electric Flux E dA Surface integral dA is 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 flux Outward E gives positive flux E dA Area dA E dA Electric Flux Example Closed cylinder of length L radius R Uniform E parallel to cylinder axis What is the total electric flux through surface of cylinder a 2 RL E b 2 R2 E c Zero R2 E R2 E 0 What goes in MUST come out Hint Surface area of sides of cylinder 2 RL Surface area of top and bottom caps each R2 dA E L dA R Electric Flux Example dA Note that E is NORMAL to both bottom and top cap E is PARALLEL to curved surface everywhere So 1 2 3 R2E 0 R2E 0 Physical interpretation total inflow total outflow 1 2 3 dA dA 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 a 4 3 R2 E 13 33 Nm2 C b 2 R2 E 20 Nm2 C c 4 R2 E 40 Nm2 C What could produce such a field What is the flux if the sphere is not centered on the charge Electric Flux Example r q E 2 r r r Inward r dA dA r q Outward r r E dA EdAcos 180 EdA Since r is Constant on the Sphere Remove E Outside the Integral r r E dA E kq 2 Surface Area Sphere dA 4 r 2 r q 4 q 0 4 0 Gauss Law Special Case Gauss Law General Case Consider any ARBITRARY CLOSED surface S NOTE this does NOT have to be a real physical object The TOTAL ELECTRIC FLUX through S is proportional to the TOTAL CHARGE ENCLOSED The results of a complicated integral is a very simple formula it avoids long calculations S r r q E dA 0 Surface One of Maxwell s 4 equations Examples r r q E dA 0 Surface Gauss Law Example Spherical symmetry Consider a POINT charge q pretend that you don t know Coulomb s Law Use Gauss Law to compute the electric field at a distance r from the charge Use symmetry draw a spherical surface of radius R centered around the charge q E has same magnitude anywhere on surface E normal to surface r q E E A E 4 r q 0 q kq E 2 2 4 0 r r 2 Gauss Law Example Cylindrical symmetry Charge of 10 C is uniformly spread over a line of length L 1 m Use Gauss Law to compute magnitude of E at a perpendicular distance of 1 mm from the center of the line Approximate as infinitely long line E radiates outwards Choose cylindrical surface of radius R length L co axial with line of charge E 1m R 1 mm Gauss Law cylindrical symmetry cont Approximate as infinitely long line E radiates outwards Choose cylindrical surface of radius R length L co axial with line of charge E 1m E A E 2 RL q L 0 0 L E 2k 2 0 RL 2 0 R R R 1 mm Compare with Example L 2 L 2 dx x E y k a 2 2 3 2 k a 2 2 2 a x a x a L 2 L 2 2k L 2 a 4a L if the line is infinitely long L a 2k L 2k Ey 2 a a L 2 Summary Electric flux a surface integral vector calculus useful visualization electric flux lines caught by the net on the surface Gauss law provides a very direct way to compute the electric flux
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