Gauss s law relates to total electric flux through a closed surface to the total enclosed charge.


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1 Chapte : Gauss s Law Gauss s Law is an altenative fomulation of the elation between an electic field and the souces of that field in tems of electic flu. lectic Flu Φ though an aea ~ Numbe of Field Lines which piece the aea depends upon geomety (oientation and size of aea, diection of ) electic field stength ( ~ density of field lines) Φ Φ cosθ θ θ pc: Gauss s law elates to total electic flu though a closed suface to the total chage. Stat with single point chage within an abitay closed suface. dd up all contibutions dφ. d Φ Φ pc:
2 intemediate steps: chage at the cente of a spheical suface two patches of aea subtending the same solid angle constant dφ dφ dding up the flu ove the suface of one of the sphees Φ k k o 4 π pc: Fo a chage in an abitay suface dφ cosθ sphee Poject aea incement onto neaest sphee : Flu though aea flu though aea incement on neaest sphee with same solid angle. Flu though neaest sphee aea incement flu though aea incement on a common sphee fo same solid angle. dd up ove all solid angles > ove entie suface of common sphee > simple sphee esults. Φ o pc: 4
3 Fo chages located outside the closed suface numbe of field lines eiting the suface (Φ ) numbe of field lines enteing the suface (Φ ) > no net contibution to Φ Gauss s Law: Φ Q o pc: 5 Using Gauss s Law Select the mathematical suface (a.k.a. Gaussian Suface)  to detemine the field at a paticula point, that point must lie on the suface  Gaussian suface need not be a eal physical suface in empty space, patially o totally embedded in a solid body Gaussian suface should have the same symmeties as chage distibution.  concentic sphee, coaial cylinde, etc. Closed Gaussian suface can be thought of as seveal sepaate aeas ove which the integal is (elatively) easy to evaluate. e.g. coaial cylinde cylinde walls caps If is pependicula to the suface ( paallel to ) and has constant magnitude then If is tangent (paallel) to the suface ( pependicula to ) then pc: 6
4 Conductos and lectic Fields in lectostatics Conductos contain chages which ae fee to move lectostatics: no chages ae moving F > fo a conducto unde static conditions, the electic field within the conducto is zeo. Fo any point within a conducto, and all Gaussian sufaces completely imbedded within the conducto within bulk conducto > all (ecess) chage lies on the suface! (fo a conducto unde static conditions) pc: 7 Conducto with void: all chage lies on oute suface unless thee is an isolated chage within void. Faaday icepail epeiment chaged conducting ball loweed to inteio of icepail ball touches pail > pat of inteio of conducto Ball comes out unchaged > veifies Gauss s Law > Coulomb s Law Moden vesions establish eponent in Coulomb s to 6 decimal places pc: 8
5 Field of a conducting sphee, with total chage and adius R R Spheical symmety > spheical Gaussian sufaces constant on suface, pependicula to suface on inteio eteio: 4π 4π R pc: 9 Field of a unifom ball of chage, with total chage and adius R R R Spheical symmety > spheical Gaussian sufaces constant on suface, pependicula to suface eteio: inteio: 4π 4π 4π 4π 4 π 4 πr R 4π R 4π R pc:
6 Line of chage (infinite), chage pe unit length λ cylindical symmety, is adially outwad (fo positive λ) Gaussian suface: finite cylinde, length l and adius l Caps: paallel to suface, Φ Cylinde: pependicula to the suface lπ lπ λ π λl pc: Symmety is the Key! Spheical Symmety Cylindical Symmety kq kλ enc enc pc:
7 pc: Field of an infinite sheet of chage, chage pe aea infinite plane, is pependicula to the plane (fo positive ) with eflection symmety Gaussian suface: finite cylinde, length centeed on plane, caps with aea Tube: paallel to suface, Φ Caps: pependicula to the sufaces pc: 4 Two oppositely chaged infinite conducting plates (/ ) plana geomety, is pependicula to the plane Gaussian sufaces: finite cylinde, length l centeed on plane, caps with aea Tube: paallel to suface, Φ Caps: pependicula to the sufaces
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