Φ E = E A E A = p212c22: 1

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Aubrey Rose Smith
5 years ago
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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 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) A Φ A A Φ A Acosθ θ θ p1c: 1

2 Gauss s law elates to total electic flux though a closed suface to the total chage. Stat with single point chage within an abitay closed suface. Add up all contibutions dφ. d Φ Φ p1c:

3 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 d constant 1 Φ Φ d Adding up the flux ove the suface of one of the sphees

4 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:

5 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

6 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

7 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

8 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

9 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

10 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

11 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 A lπ lπ λ π λl p1c: 11

12 Symmety is the Key! Spheical Symmety Cylindical Symmety kq kλ enc enc p1c: 1

13 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

14 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

2 E. on each of these two surfaces. r r r r. Q E E ε. 2 2 Qencl encl right left 0

2 E. on each of these two surfaces. r r r r. Q E E ε. 2 2 Qencl encl right left 0 Ch : 4, 9,, 9,,, 4, 9,, 4, 8 4 (a) Fom the diagam in the textbook, we see that the flux outwad though the hemispheical suface is the same as the flux inwad though the cicula suface base of the hemisphee