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lecture05

# Fundamentals of Physics, (Chapters 21- 44) (Volume 2)

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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 Consider an airstream of velocity that is aimed at a loop of area . The velocity vector is at angle with respect to the ˆ loop normal . The product cos is know n as v A v n vA ! ! " = Flux of a Vector. r r the . 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 vA v ! ! " = flux Note 1: r plane). 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 magnitude equal to . v vA v A A A ! ! = ° = # Note 2 : r r r r (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 ! 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 a A E A EA E A " # \$# = ! = \$# % r r r r 2 Flux SI unit: N m / C s the area 0. The limit of the sum becomes the integral: The circle on the integral sign indicates that the surface is closed. When we apply Gauss' la A E dA ! = \$ # \$ & Note 1: r r ° w the surface is known as "Gaussian." is proportional to the net number of electric field lines that pass through the surface. ! Note 2 : E dA ! = " # r r ° (23-3)

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0 enc 0 enc The flux of through any closed surface net charge enclosed by the surface. Gauss' law can be formulated as follows: In equ ation form: Equivalently: E q q ! ! " = # = Gauss' Law r 0 enc å E dA q \$ = % r r ° 0 enc q ! " = 0 enc å E dA q ! = " r r ° 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 : Examp 1 0 1 2 0 2 3 0 3 4 0 4 1 2 3 4 Surface : , Surface : Surface : 0, Surface : 0 We refer to , , , as "Gaussian surfaces." S q S q S S q q S S S S ! ! ! ! " = + " = # " = " = # + = le : Note : (23-4)
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 ' law and vice versa.

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