Phy 213: General Physics III

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Malcolm Hicks
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1 Phy 1: Geneal Physics III Chapte : Gauss Law Lectue Ntes E Electic Flux 1. Cnside a electic field passing thugh a flat egin in space w/ aea=a. The aea vect ( A ) with a magnitude f A and is diected nmal t the suface. The electic flux thugh A, Φ E, the uantity f electic field thugh that egin: Φ = E A = E A csφ E A 4 A 6 A φ E A A avg A 5. Cnceptually, the electic flux epesents the flw f electic field lines (nmal) thugh a egin f space. In geneal, the electic flux thugh a egin f n aeas, A i,, is the sum f the fluxes thugh all f thse egins: Φ = Φ + Φ Φ = E A E 1 i i i i A 1 E 1

2 Gauss Law 1. Gauss Law is a fundamental cnsevatin law f natue elating electic chage t electic flux. Accding t Gauss Law, the ttal electic flux thugh any clsed ( Gaussian ) suface is eual t the enclsed chage (Q enclsed ) divided by the pemittivity f fee space (ε ): enclsed ΦE= Ei A= i i ε O in geneal, enclsed ΦE= E da = ε. Gauss Law can be used t detemine the electic field (E) f many physical ientatins ( distibutins) f chage 4. Many cnseuences f Gauss Law pvide insights that ae nt necessaily bvius when applying Culmb s Law Gauss Law f a Pint Chage Cnside a pint chage, Gaussian sphee 1. An apppiate gaussian suface f a pint chage is a cncentic sphee, adius, since all E fields will be t the suface. The electic flux thugh the suface is: Φ = E da = E da = E A = E 4 π E clsed suface. Applying Gauss Law yields E: Φ = E( 4π ) = E = ε 4 πε sphee ( ) enclsed enclsed E

3 Spheical Symmety: Chaged Sphee 1. Spheical symmety can be explited f any spheical shape. Cnside a hllw cnducting sphee, adius R & chage : R Gaussian sphee. The flux thugh a gaussian sphee is: enclsed πε Φ = E A = E 4 E ( π ) 4. Applying Gauss Law: =E ( 4 ) = E= Φ π 4 enclsed 5. F > R: E = 4πε 6. F < R: E= = 0 { enclsed= 0C} Inside a hllw 4 cnduct, E = 0 enclsed enclsed E ε πε Spheical Symmety: Chaged Sphee () Cnside a slid nn-cnducting sphee with adius R & unifm chage density, ρ: 1. Applying Gauss Law:. F > R:. F < R: R ρ Gaussian sphee =E 4 = E= Φ 4 ( π ) ( πr ) enclsed enclsed E ε πε 4 ρ enclsed ρr πε πε ε 4 ρ( π ) enclsed ρ πε πε ε E = = = 4 4 E = = = 4 4 Inside a unifmly chaged insulat, E 0

4 Spheical Symmety: Chaged Sphees () Cnside cncentic hllw cnducting sphees with adii R 1 & R 1 and chages 1 & : Gaussian sphee 1 R 1 R 1. Applying Gauss Law: =E ( 4 ) = enclsed E= enclsed ΦE π ε 4 πε enclsed. F < R 1 : E = = 0 4πε enclsed 1. F R > > R 1 : E = = 4πε 4πε ( 1+ enclsed ) 4. F > R : E = = {ppsite chage subtacts} 4πε 4πε Applicatin: Line f Chage 1. Cylindical symmety is useful f evaluating a line f chage as well as chaged cylindical shapes. Cnside an infinitely lng line f chage, chage density λ: λ The flux thugh each f the sufaces is: Φ = Φ + Φ + Φ = Φ since Φ = Φ = 0 Φ 4. Applying Gauss Law: E 1 1 enclsed x E=E( π) x= = λ ε ε x λ E = πε 4

5 Plana Symmety: A Chaged Sheet 1. A Gaussian pill-bx is an apppiate Gaussian suface f evaluating a chaged flat sheet. Cnside a unifmly chaged flat sheet with a suface chage density, σ: Gaussian pill-bx. The flux thugh the pill-bx suface is: Φ = Φ + Φ + Φ = Φ { Φ Φ Φ } E side face1 face face = = face face1 face 4. The electic field: enclsed enclsed σπ σ ΦE= E ( π ) = E= = = ε πε πε ε Applicatin: Paallel Chaged Plates 1. Tw ppsitely chaged cnducting plates, with chage density σ:. Suface chages daw twad each the n the inne face f each plate. T detemine the electic field within the plates, apply a Gaussian pill-bx t ne f the plates 4. The flux thugh the pill-bx suface is: ΦE= Φside+ Φinne face+ Φute face= Φute face { whee Φ = Φ =0} inne face side 5. The electic field is: enclsed enclsed σπ σ ΦE=E ( π ) = E= = = ε πε πε ε + Gaussian pill-bx - 5

6 Cnseuences f Gauss Law 1. Electic flux is a cnseved uantity f an enclsed electic chage. The electic field inside a chaged slid cnduct is ze. The electic field inside a chaged hllw cnduct nn-cnduct is ze 4. Gemetical symmeties can make the calculatin f an electic field less cumbesme even f cmplicated chage distibutins 6

CHAPTER 24 GAUSS LAW

CHAPTER 24 GAUSS LAW CHAPTR 4 GAUSS LAW LCTRIC FLUX lectic flux is a measue f the numbe f electic filed lines penetating sme suface in a diectin pependicula t that suface. Φ = A = A csθ with θ is the angle between the and