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Unformatted text preview: Chapter 23 Gauss’ Law • The new concept: flux of the electric field ( symbol Φ ) • Introduce the Gauss’ law and apply it to determine the electric filed generated by – A point charge – An infinite, uniformly charged insulating plane 1 – An infinite, uniformly charged insulating rod – A uniformly charged spherical shell – A uniform spherical charge distribution • Apply Gauss’ law to determine the electric field inside and outside charged conductors • Introduce area vector A which has the magnitude equal to A and the direction of the loop normal. Then • Note: Flux depends on θ A v θ vA r r ⋅ = = cos Φ Flux n ˆ • Consider an airstream of velocity v which is aimed at a loop of area A . The velocity vector v is at angle θ with respect to the loop normal . The product vA cos θ is known as the flux Φ . (In this example the flux is equal to the volume flow rate through the loop) ˆ n 2 ˆ n – It is maximal and equals to vA when θ = – It is minimal and equals to when θ = 90 • Consider the closed surface shown in the figure. Assume we know the electric field E in the vicinity of the surface. We define flux Φ of the electric field through the surface as follows: Divide the surface into small “elements” of area ∆ A or each element calculate the flux Flux of an Electric Field 3 For each element calculate the flux  Then calculate the sum  Let element ∆ A be infinitely small so that the sum becomes an integral θ A E A E cos ∆ ⋅ ∆ ⋅ = ⋅ r r ∑ ⋅ = A E r r ∆ Φ ∫ ⋅ = A d E r r Φ SI units: N·m 2 /C • Electric field flux Φ through a closed surface is given by • The loop on the integral indicates that integration surface is closed. Such closed surface is called ∫ ⋅ = A d E r r Φ Flux of an Electric Field (cont’d) 4 “ Gaussian surface ” • Since magnitude of the electric field is proportional to the density of the field lines, we can conclude that: The electric field flux Φ through a Gaussian surface is proportional to the net number of electric field lines passing through that surface • Gauss’ law relates the net flux Φ of an electric field through a closed surface (a Gaussian surface) to the net charge q enc that is enclosed by that surface • In equation form: auss’s law holds for ny losed surface. In practice a particular enc q A d E ε = ⋅ ∫ r r or equivalently enc q ε = Φ Gauss’ Law 5 ˆ n ˆ n ˆ n • Gauss’s law holds for any closed surface. In practice a particular surface makes the problem of determining the field very simple • When calculating the net charge inside a closed surface we take into account the algebraic signs of all charges enclosed • If enclosed charge q enc is positive, the net electric flux is outward; if enclosed charge q enc is negative, then the net flux is inward • When applying Gauss’ law for a closed surface we ignore the charges outside the surface (no matter how large they are) • Example : Calculate the net flux through...
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 Fall '08
 IASHVILI
 Charge, Electric charge, Charged Conductor

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