Physics II chapter 22 notes

Physics II chapter 22 notes - Chapter 22 Gausss Law...

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V6.03 BQ/RS F’10 19 Chapter 22: Gauss’s Law Gauss’s law is an alternative to Coulomb’s law (more general, in fact!) Introduction In situations of high symmetry, can greatly simplify the determination of E If E is known, Gauss’s law gives information about source charge distribution Important ingredient: the concept of electric flux Relates the electric flux through a closed surface to the enclosed charge First, consider the flow of some fluid (water, maple syrup, beer,…) through a Electric Flux (22-2) small dx dy × square loop of dimension in the xy -plane as shown Let dV be volume of fluid flowing upward through loop during small time dt If fluid flows with velocity ˆ vk then ( ) ( ) dV v dt dx dy = If fluid flows with velocity ˆ vj then 0 dV = If fluid flows with velocity ˆ ˆ cos sin v kv j θθ + (still speed v ) then only upward motion (in +z direction) through loop counts speed component in x or y direction does not contribute Area of loop dA dx dy = , vector normal to its surface ˆ ˆ nk = Define infinitesimal surface area element by ˆ ˆ ( ) k dx dy dA n dA = = From above examples, we see that x y z dy dx v v dt x y z dy dx v θ v dt z v dt ( ) ( ) ( ) ( ) cos z dV v dt dx dy v dt dx dy = = x y z dy dx v dV v dA dt =
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V6.03 BQ/RS F’10 20 Rate dA of volume flow of fluid (flux) through loop is By analogy, define electric flux through infinitesimal surface area element dA as Note: nothing is actually flowing (treat field lines like lines of flow) Not well defined until we decide which of two possible ˆ n directions to use for dA the choice of the direction of dA defines positive flux if flux is in one direction, then flux in opposite direction is −Φ Electric flux through an arbitrary surface S obtained by summing contributions from the infinitesimal surface area elements making up the surface note: dA must everywhere point to the same side of surface electric flux is a scalar quantity which may be positive or negative electric flux has units of 2 Nm /C for a closed surface, we usually choose dA to be outward semantics for closed surfaces: total outward flux means total flux through surface defining positive as the outward direction pointing which gives us the flux coming out of the closed surface positive inward flux is equivalent to negative outward flux both E and dA (and the angle between them) may vary with location on surface E Φ depends on values of E only on the surface S dV v dA dt = E d E dA Φ= E S E dA
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V6.03 BQ/RS F’10 21 Example: electric flux through flat square surface due to point charge point charge q + at origin aa × square loop distance a away, parallel to xz -plane, located as shown Electric field from charge Express in terms of ,, xyz Electric field Infinitesimal surface area element ˆ dA dx dz j =
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This note was uploaded on 08/25/2011 for the course PHYSICS II 33-107 taught by Professor B.quinn during the Summer '10 term at Carnegie Mellon.

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Physics II chapter 22 notes - Chapter 22 Gausss Law...

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