Lecture 5 Lecture 5 --GaussGauss’’LawLawChapter 27 Chapter 27 --Tuesday January 23rdTuesday January 23rd•Review of Thursday’s class•The flux of a vector field•The flux of an electric field•Applications of Gauss’ law (examples)•Gauss’ Law and conductors•Experimental tests of Gauss’ law •Some sophisticated vector calculus Reading: pages 612 thru 625 (chapter 27) in HRKReading: pages 612 thru 625 (chapter 27) in HRKRead and understand the sample problemsRead and understand the sample problemsWebAssign: set 2, due Thu. 25th at 11:59pmWebAssign: set 2, due Thu. 25th at 11:59pmGraded problems (Ch. 27) Graded problems (Ch. 27) ––Ex. 15, 18, 21; Prob. 2, 10, 18Ex. 15, 18, 21; Prob. 2, 10, 18Practice problems (Ch.27): Ex. 21,25,27,29; Prob. 11,17Practice problems (Ch.27): Ex. 21,25,27,29; Prob. 11,17••Exam 1 is two weeks from today (Feb 6th).Exam 1 is two weeks from today (Feb 6th).The flux of a vector fieldThe flux of a vector fieldWhat if there are multiple What if there are multiple surface elements to consider? surface elements to consider? Then,.Φ= ⋅∑vAGGFor a closed surface, we ALWAYS choose to point outwards. This is very important for Gauss’ Law!!AGsign now determinedabcΦ= ⋅∑vAGGThe flux through a closed curved surfaceThe flux through a closed curved surfacedΦ= ⋅∫vAGGvThe flux of an electric fieldThe flux of an electric fieldEdΦ=⋅∫EAGGvGaussGauss’’law is concerned with the flux of E through closed surfaceslaw is concerned with the flux of E through closed surfaces••You may recall that when we developed our graphical You may recall that when we developed our graphical reprerepre--sentationsentationof electric field lines, the electric field strength was of electric field lines, the electric field strength was proportional to the number of field lines crossing a unit area proportional to the number of field lines crossing a unit area perpendicular to the field.perpendicular to the field.••Consequently,Consequently,()##.of field linesFlux area of field linesarea=×⊥=⊥∑∑••In other words, the flux of E through a surface is proportional In other words, the flux of E through a surface is proportional to the number of field lines penetrating the surface.to the number of field lines penetrating the surface.••This is the essence of GaussThis is the essence of Gauss’’law.law.••Recall also that the number of field lines is related to the Recall also that the number of field lines is related to the number of charges producing the electric field.number of charges producing the electric field.The flux of an electric fieldThe flux of an electric fieldEdΦ=⋅∫EAGGvΦE= 01E enclosedodqεΦ= ⋅ =∑∫EAGGvGaussGauss’’law is concerned with the flux of E through closed surfaceslaw is concerned with the flux of E through closed surfacesCharge densitiesCharge densitiesIn 1D (a line or wire):In 1D (a line or wire):,orQdQLdLλλ==λis the line charge density, or charge per unit length, in Coulombs per meter. L represents length, and Q is charge.In 2D (a surface or sheet):In 2D (a surface or sheet):,orQdQAdAσσ==σis the surface charge density, or charge per unit area in Coulombs per meter2; A represents area, and Q is charge.In 3D (a solid object):In 3D (a solid object):,orQdQVdVρρ==ρis the volume charge density, or charge per unit volume in Coulombs per meter3. V represents volume, and Q is charge.GaussGauss’’law and conductorslaw and conductors••The electric field inside The electric field inside a conductor which is in a conductor which is in electrostatic electrostatic equilibrium must be equilibrium must be zero. zero. ••Equilibrium is reached Equilibrium is reached very quickly very quickly (<10(<10--9 9 s).s).GaussGauss’’law and conductorslaw and conductorsAn excess charge An excess charge placed on an placed on an isolated conductor isolated conductor moves entirely to moves entirely to the outer surface of the outer surface of the conductor. None the conductor. None of the excess charge of the excess charge is found within the is found within the body of the body of the conductor.conductor.GaussGauss’’law and conductorslaw and conductors••No charge resides on No charge resides on any inside surface, any inside surface, unless...unless...GaussGauss’’law and conductorslaw and conductors••No charge resides on No charge resides on any inside surface, any inside surface, unless...unless...••...a charge is placed ...a charge is placed within the cavity.within the cavity.••In this situation, In this situation, charge within the charge within the conductor moves so as conductor moves so as to screen the field set to screen the field set up by the charge in the up by the charge in the cavity. cavity. +q-------+++++++E = 0The electric field outside a conductorThe electric field outside a conductorEsurfaceouter cap inner cap sidewallsddddΦ= ⋅=⋅+⋅+⋅∫∫∫∫EAEA EA EAGGGGGGG GvvvvoEσε=The electric field outside a conductorThe electric field outside a conductor••At first sight, the At first sight, the result from the result from the previous slide previous slide seems odd in view seems odd in view of the earlier result of the earlier result for a sheet of for a sheet of charge.charge.••However, one must However, one must not forget that the not forget that the field is due not only field is due not only to the charge at the to the charge at the immediate surface, immediate surface, but also from the but also from the charge elsewhere charge elsewhere on the surface of the on the surface of the conductor.conductor.The electric field outside a conductorThe electric field outside a conductor••If the conductor is If the conductor is completely isolated completely isolated and has a uniform and has a uniform shape (e.g. sphere, shape (e.g. sphere, sheet, cylinder), sheet, cylinder), then the charge then the charge distributes evenly distributes evenly over the surface.over the surface.••If the shape is nonIf the shape is non--uniform (e.g. heart uniform (e.g. heart shape), or it is not shape), or it is not isolated, then the isolated, then the charge may be charge may be distributed highly distributed highly nonnon--uniformly.uniformly.The electric field outside a conductorThe electric field outside a conductorThe charge on a parallel plate The charge on a parallel plate capacitor resides only on the capacitor resides only on the inner
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