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HR23_post - Chapter 23 Gauss Law The new concept flux of...

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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

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Introduce area vector A which has the magnitude equal to A and the direction of the loop normal. Then Note: Flux depends on θ – It is maximal and equals to vA when θ = 0 0 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 minimal and equals to 0 when θ = 90 0
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 - For each element calculate the flux Flux of an Electric Field 3 - 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

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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
ˆ n 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: Gauss’s law holds for any closed surface. In practice a particular enc q A d E ε = r r 0 or equivalently enc q ε = Φ 0 Gauss’ Law 5 ˆ n ˆ n 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)

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ˆ n Example : Calculate the net flux through surfaces S 1 , S 2 , S 3 and S 4 , (Fig. on the left) Surface S 1 : ε 0 Φ 1 = +q Surface S 2 : ε 0 Φ 2 = –q ε Φ enc q ε = Φ 0 Gauss’ Law (cont’d) 6 ˆ n
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