Physics 2102 Jonathan Dowling Flux Capacitor Operationa Physics 2102 Lecture 07 WED 28 JAN Gauss Law III QuickTime and a decompressor are needed to see this Version 1 22 07 Carl Friedrich Gauss 1777 1855 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 S r r q E dA 0 Surface The results of a complicated integral is a very simple formula it avoids long calculations One of Maxwell s 4 equations Examples r r q E dA 0 Surface Gauss Law 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 E R 1 mm 1m Gauss Law Cylindrical Symmetry Approximate as infinitely long line E radiates outwards Choose cylindrical surface of radius R length L co axial with line of charge E R 1 mm 1m E A E 2 RL q L 0 0 L E 2k 2 0RL 2 0R R Recall Finite Line 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 2 a 4a L the Line Is Infinitely Long L a 2k L 2k Ey 2 a a L r E QuickTime and a TIFF Uncompressed decompressor are needed to see this picture a R Gauss Law Insulating Plate Infinite INSULATING plane with uniform charge density s E is NORMAL to plane Construct Gaussian box as shown q A Applying Gauss law we have 2 AE 0 0 Solving for the electric field we get E 2 0 For an insulator E 2 0 and for a conductor E 0 Insulating and Conducting Planes Q E 2 0 2 A 0 Q Insulating Plate Charge Distributed Homogeneously Q 2 Q E 0 2 A 0 Conducting Plate Charge Distributed on the Outer Surfaces Gauss Law 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 place spherical surface of radius R centered around the charge q E has same magnitude anywhere on surface E normal to surface q 0 r q E E A E 4 r q kq E 2 2 4 0 r r 2 Electric Fields With Spherical Symmetry Shell Theorem 10 C A spherical shell has a charge of 10C and a point charge of 15C at the center What is the electric field produced OUTSIDE the shell 15C If the shell is conducting And if the shell is insulating Charged Shells Behave Like a Point Charge of Total Charge Q at the Center E E k 15C r2 E 0 E k 5C r2 r Conducting Electric Fields With Spherical Symmetry Insulating QuickTime and a TIFF Uncompressed decompressor are needed to see this picture 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 QuickTime and a decompressor are needed to see this picture
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