Chapter_22 - Chapter 22: Gauss's Law Introduction Gauss's...

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15 Chapter 22: Gauss’s Law Introduction ± Gauss’s law is an alternative to Coulomb’s law (more general, in fact!) ± In situations of high symmetry, can greatly simplify the determination of E G ± If E G 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 Electric Flux (22-2) ± Consider the flow of some fluid (water, maple syrup, ale,…) through a small square loop of dimension dx dy × 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 vdt dxdy = If fluid flows with velocity ˆ vj then 0 dV = If fluid flows with velocity ˆ ˆ cos sin vk vj θ + (still speed v ) then only upward motion (in +z direction) through loop counts speed component in 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 dA n dA k dx dy == G From above examples, we see that x y z dy dx v vdt x y z dy dx v z vd t () ( ) ( ) ( ) cos z dV v dt dx dy t dxdy x y z dy dx v dV v dA dt =⋅ G G
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16 ± Rate of volume flow of fluid (flux) through dA G loop is ± By analogy, define electric flux through infinitesimal surface area element dA G as note that nothing is actually flowing (treat field lines like lines of flow) ± Electric flux through an arbitrary surface S obtained by summing contributions from the infinitesimal surface area elements making up the surface note: dA G must everywhere point to the same side of surface one can pass through a surface in one of two directions ¾ electric flux through a surface can only be defined if you specify which of the two directions through the surface you accept as the direction of positive flux ¾ the choice of the direction of dA G defines positive flux ¾ if flux is in one direction, then flux in opposite direction is −Φ 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 G to be outward pointing which gives us the flux coming out of the closed surface semantics for closed surfaces: total outward flux means total flux through surface defining positive as the outward direction ¾ positive inward flux is equivalent to negative outward flux remember that both E G and dA G may vary with location on surface E Φ depends on values of E G only on the surface S dV vdA dt = G G E dE d A Φ= G G E S EdA G G
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17 ± 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 = G Electric flux (rightward) calculated from surface integral ±
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Chapter_22 - Chapter 22: Gauss's Law Introduction Gauss's...

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