Physics 2102 Jonathan Dowling Flux Capacitor Operational Physics 2102 Lecture 4 Gauss Law II Version 1 23 07 Carl Friedrich Gauss 1777 1855 HW and Exam Solutions www phys lsu edu classes spring2007 phys2102 Solutions index html USERNAME Phys2102 Password Solution1 Both are cAsE SenSiTivE Properties of conductors Inside 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 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 R2 R1 15 C 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 Since E 0 inside the metal flux through this surface 0 Gauss Law says total charge enclosed 0 Charge on inner surface 15 C 5 C Since TOTAL charge on shell is 10 C Charge on outer surface 10 C 15 C 5 C 15C 15C Faraday 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 0 Choose any arbitrary surface inside the metal Since E 0 flux 0 Hence total charge enclosed 0 All charge goes on outer surface Inside cavity is shielded from all external electric fields Faraday Cage effect More Properties of conductors We 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 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 Infinite INSULATING plane with uniform charge density E is NORMAL to plane Construct Gaussian box as shown Applying Gauss law q A we have 2 AE 0 0 Solving for the electric field we get E 2 0 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 conductor A ApplyingGauss law wehave AE 0 Solving for the electric field we get E 0 For 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 planes Q E 2 0 2 A 0 Q Insulating plate charge distributed homogeneously Q 2 Q E 0 2 A 0 Conducting plate charge distributed on the outer surfaces 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 E 0 A ApplyingGauss law wehave AE 0 Solving for the electric field we get E 0 THIS IS A GENERAL RESULT FOR CONDUCTORS Electric fields with spherical symmetry shell theorem 10 C A 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 15C If the shell is conducting And if the shell is insulating Charged Shells Behave Like a Point Charge of Total Charge Q at the Center E E k 15C r2 E 0 E k 5C r2 r Conducting 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 E 0
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