# HR23 - Chapter 23 Gauss law In this chapter we will...

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Chapter 23 Gauss’ law In this chapter we will introduce the following new concepts: -The flux (symbol Φ ) of the electric field. -Symmetry Introduce: Gauss’ law We will then apply Gauss’ law and determine the electric field generated by: An infinite, uniformly charged insulating plane. An infinite, uniformly charged insulating rod. A uniformly charged spherical shell. A uniform spherical charge distribution. We will also apply Gauss’ law to determine the electric field inside and outside charged conductors. (23-1)

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ˆ n ˆ n Consider an air stream of velocity which is aimed at a loop of area A . The velocity vector is at angle with respect to the ˆ loop normal . The product cos known as t v v n vA θ θ Φ = Flux of a vector. r r he . In this example the flux is equal to the volume flow rate through the loop (thus the name flux) depends on . It is maxi dV dt θ Φ flux Note 1: mum and equal to for 0 ( perpendicular to the loop plane). It is minimum and equal to zero for 90 . ( parallel to the loop plane). cos The vector is parallel t vA v v vA v A A θ θ θ = = ° = Note 2 : r r r r r o the loop normal and has magnitute equal to . A (23-2)
ˆ n ˆ n ˆ n Consider the closed surface shown in the figure. In the vicinity of the surface assume that we have a known electric field . The flux of the electric field thro h ug E Φ Flux of the electric field. r the surface is defined as follows: Divide the surface into small " " of area For each element calculate the term cos Form the sum Take the limit of the sum as A E A E A E A θ ⋅∆ = Φ = ⋅∆ 1. 2. 3. 4. r r r r elements 2 Flux SI unit: N m / the area 0 The limit of the sum becomes the integral: The circle on the intergral sign indicates that the integration surface is closed. When we appl dA C A E Φ = Note 1: r r Ñ y Gauss' law this surface is known as "Gaussian surface" is proportinal to the net number of electric field lines that pass through the surface Φ Note 2 : E dA Φ = r r Ñ (23-3)

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ˆ n ˆ n ˆ n Gauss' law can be formulated as follows: In equation form: Equivalently: enc o enc o E q E ε = Φ = o The flux of through any closed surface×ε net charge enclosed by the sur Gauss' L e aw fac r r enc dA q = r Ñ o enc q ε Φ = o enc ε E dA q = r r Ñ Gauss' law holds for closed surface. Usually one particular surface makes the problem of determining the electric field very simple. When calculating the net charge inside a c Note 1: any Note 2 : losed surface we take into account the algebraic sign of each charge When applying Gauss' law for a closed surface we the charges the surface no matter how large they are. Note 3 : ignore outside Exampl 1 1 2 2 3 3 4 4 1 2 3 4 Surface S : , Surface S : Surface S : 0 , Surface S : 0 We refer to S , S , S , S as "Gaussian surfaces" o o o o q q q q ε ε ε ε Φ = + Φ = - Φ = Φ = - + = e : Note : (23-4)
Karl Friedrich Gauss 1777-1855

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ˆ n dA Gauss' law and Coulomb's law are different ways of describing the relation between electric charge and electric field in static cases. One can derive Coulomb's law from Gauss Gauss' law and Coulomb's law ' laws and vice versa.
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