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 flux. lectic Flux Φ though an aea A ~ 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) http://physp.sl.psu.edu/phys_anim/m/flux.avi A Φ A A Φ A Acosθ θ θ p1c: 1
Gauss s law elates to total electic flux though a closed suface to the total chage. http://physp.sl.psu.edu/phys_anim/m/gauss.html Stat with single point chage within an abitay closed suface. Add up all contibutions dφ. d Φ Φ p1c:
p1c: o k k π Φ intemediate steps: chage at the cente of a spheical suface two patches of aea subtending the same solid angle 1 1 1 d constant 1 Φ Φ d Adding up the flux ove the suface of one of the sphees
Fo a chage in an abitay suface dφ cosθ sphee Poject aea incement onto neaest sphee : Flux though aea flux though aea incement on neaest sphee with same solid angle. Flux though neaest sphee aea incement flux though aea incement on a common sphee fo same solid angle. Add up ove all solid angles > ove entie suface of common sphee > simple sphee esults. Φ o p1c:
Fo chages located outside the closed suface numbe of field lines exiting the suface (Φ ) numbe of field lines enteing the suface (Φ ) > no net contibution to Φ Gauss s Law: Φ Q o p1c: 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, coaxial cylinde, etc. Closed Gaussian suface can be thought of as seveal sepaate aeas ove which the integal is (elatively) easy to evaluate. -e.g. coaxial cylinde cylinde walls caps If is pependicula to the suface ( paallel to ) and has constant magnitude then A If is tangent (paallel) to the suface ( pependicula to ) then p1c: 6
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 (excess) chage lies on the suface! (fo a conducto unde static conditions) p1c: 7
Conducto with void: all chage lies on oute suface unless thee is an isolated chage within void. Faaday ice-pail expeiment 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 exponent in Coulomb s to 16 decimal places p1c: 8
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 exteio: A π π R p1c: 9
p1c: 1 Field of a unifom ball of chage, with total chage and adius R Spheical symmety > spheical Gaussian sufaces constant on suface, pependicula to suface exteio: inteio: A π π R R R R R R A π π π π π π
Line of chage (infinite), chage pe unit length λ http://physp.sl.psu.edu/phys_anim/m/gauss_line.avi 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 A lπ lπ λ π λl p1c: 11
Symmety is the Key! Spheical Symmety Cylindical Symmety kq kλ enc enc p1c: 1
Field of an infinite sheet of chage, chage pe aea σ infinite plane, is pependicula to the plane (fo positive σ) with eflection symmety Gaussian suface: finite cylinde, length x centeed on plane, caps with aea A Tube: paallel to suface, Φ A x x Caps: pependicula to the sufaces x A x σ σa x A p1c: 1
p1c: 1 Two oppositely chaged infinite conducting plates (/ σ) plana geomety, is pependicula to the plane Gaussian sufaces: finite cylinde, length l centeed on plane, caps with aea A Tube: paallel to suface, Φ Caps: pependicula to the sufaces σ σ x x x A A A