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 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 airstream 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 a t s h v v n vA θ Φ = Flux of a vector. r r e . In this example the flux is equal to the volume flow rate through the loop (thus the name flux) depends on . It is maximum and equal to for 0 ( perpendicular to the loop pl vA v Φ = flux Note 1: r ane). It is minimum and equal to zero for 90 . ( parallel to the loop plane). cos The vector is parallel to the loop normal and has magnitute equal to . v vA v A A A = ° = ⋅ Note 2 : r r r r (23-2)
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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: 1. Divide the surface into small "elements" of area 2. For each element calculate the term cos 3. Form the sum 4. Take the limit of the sum as t A E A EA E A θ ⋅∆ = Φ = ⋅∆ r r r r 2 Flux SI unit: he area 0 The limit of the sum becomes the integral: The circle on the intergral sign indicates that the closed surface is closed. When we appl N G m / y aus C d A E A Φ = ∆ → Note 1: r r Ñ s' law the surface is known as "Gaussian" 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 The flux of through any closed surface net charge enclosed by the surface Gauss' law can be formulated as follows: In equation form: Equivalently: o enc o enc E q ε × = Φ = Gauss' Law r o enc E dA q = r r Ñ o enc q εΦ = 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 ignore the charges outside the surface no matter how large they are . Note 3 : 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 Φ = + Φ = - Φ = Φ = - + = e : Note : (23-4)
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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
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This note was uploaded on 11/15/2010 for the course PHYSICS 1322 taught by Professor Michaelgorman during the Spring '10 term at University of Houston.

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

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