# 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 ice-pail epeiment chaged conducting ball loweed to inteio of ice-pail 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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