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LSU PHYS 2102 - Gauss’ Law II

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Physics 2102 Lecture 4HW and Exam SolutionsProperties of conductorsGauss’ Law: ExampleSlide 5Faraday’s CageMore Properties of conductorsCharges in conductorsSlide 9Slide 10Insulating and conducting planesSlide 12Electric fields with spherical symmetry: shell theoremSummary:PowerPoint PresentationPhysics 2102 Physics 2102 Lecture 4Lecture 4Gauss’ Law IIGauss’ Law IIPhysics 2102Jonathan DowlingCarl Friedrich Gauss 1777-1855Version: 1/23/07Flux Capacitor (Operational)HW and Exam Solutions•www.phys.lsu.edu/classes/spring2007/phys2102/Solutions/index.html•USERNAME: Phys2102 •Password: Solution1 •Both are: cAsE SenSiTivE!Properties of conductorsProperties of conductorsInside a conductor in electrostatic equilibrium, the electric field is ZERO. Why?Because if the field is not zero, then charges inside the conductor would be moving.SO: charges in a conductor redistribute themselves wherever they are needed to make the field inside the conductor ZERO. Excess charges are always on the surface of the conductors.Gauss’ Law: Example Gauss’ Law: Example • A spherical conducting shell has an excess charge of +10 C. • A point charge of 15 C is located at center of the sphere.•Use Gauss’ Law to calculate the charge on inner and outer surface of sphere(a) Inner: +15 C; outer: 0(b) Inner: 0; outer: +10 C(c) Inner: +15 C; outer: -5 C-15 CR2R1Gauss’ Law: Example Gauss’ Law: Example • Inside a conductor, E = 0 under static equilibrium! Otherwise electrons would keep moving!• Construct a Gaussian surface inside the metal as shown. (Does not have to be spherical!)-15C•Since E = 0 inside the metal, flux through this surface = 0• Gauss’ Law says total charge enclosed = 0 • Charge on inner surface = +15 CSince TOTAL charge on shell is +10 C,Charge on outer surface = +10 C  15 C = 5 C!-5 C+15CFaraday’s CageFaraday’s Cage•Given a hollow conductor of arbitrary shape. Suppose an excess charge Q is placed on this conductor. Suppose the conductor is placed in an external electric field. How does the charge distribute itself on outer and inner surfaces?(a) Inner: Q/2; outer: Q/2(b) Inner: 0; outer: Q(c) Inner: Q; outer: 0Inside cavity is “shielded” from all external electric fields! “Faraday Cage effect”• Choose any arbitrary surface inside the metal• Since E = 0, flux = 0• Hence total charge enclosed = 0• All charge goes on outer surface!More Properties of conductorsMore Properties of conductorsWe know the field inside the conductor is zero, and the excess charges are all on the surface. The charges produce an electric field outside the conductor. On the surface of conductors in electrostatic equilibrium, the electric field is always perpendicular to the surface. Why?Because if not, charges on the surface of the conductors would move with the electric field.Charges in conductorsCharges in conductors•Consider a conducting shell, and a negative charge inside the shell. •Charges will be “induced” in the conductor to make the field inside the conductor zero.•Outside the shell, the field is the same as the field produced by a charge at the center!Gauss’ Law: Example Gauss’ Law: Example •Infinite INSULATING plane with uniform charge density •E is NORMAL to plane•Construct Gaussian box as shownAEA2 have, we,q law Gauss' Applying00=Φ=εσε02get wefield, electric for the Solvingεσ=EGauss’ Law: Example Gauss’ Law: Example •Infinite CONDUCTING plane with uniform areal charge density •E is NORMAL to plane•Construct Gaussian box as shown.•Note that E = 0 inside conductorAEA=0 have, welaw, Gauss' Applyingεσ0get wefield, electric for the Solvingεσ=EFor an insulator, E=/20, and for a conductor, E=/0. Does the charge in an insulator produce a weaker field than in a conductor?Insulating and conducting Insulating and conducting planesplanesQ0022 εεσAQE ==Insulating plate: charge distributed homogeneously.002 εεσAQE ==Conducting plate: charge distributed on the outer surfaces.Q/2Gauss’ Law: Example Gauss’ Law: Example •Charged conductor of arbitrary shape: no symmetry; non-uniform charge density•What is the electric field near the surface where the local charge density is ?(a) 0(b) Zero(c)  AEA=0 have, welaw, Gauss' Applyingεσ0get wefield, electric for the Solvingεσ=EE=0++++++++++++THIS IS AGENERALRESULT FORCONDUCTORS!Electric fields with spherical Electric fields with spherical symmetry: shell theoremsymmetry: shell theorem-15C+10 CA spherical shell has a charge of +10C and a point charge of –15C at the center.What is the electric field produced OUTSIDE the shell?If the shell is conducting:ErE=k(15C)/r2E=k(5C)/r2E=0And if the shell is insulating?Charged Shells Behave Like a Point Charge of Total Charge “Q” at the CenterOnce Outside the Last Shell!Charged Shells Behave Like a Point Charge of Total Charge “Q” at the CenterOnce Outside the Last Shell!ConductingSummary:Summary:•Gauss’ law provides a very direct way to compute the electric flux.•In situations with symmetry, knowing the flux allows to compute the fields reasonably easily.•Field of an insulating plate: /20, ; of a conducting plate: /0.. •Properties of conductors: field inside is zero; excess charges are always on the surface; field on the surface is perpendicular and


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