Physics 202, Lecture 4. Gauss s Law: Review

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1 Physics 202, Lectue 4 Today s Topics Review: Gauss s Law Electic Potential (Ch. 25-Pat I) Electic Potential Enegy and Electic Potential Electic Potential and Electic Field Next Tuesday: Electic Potential (Ch. 25-Pat II) Homewok #1 due tomoow (9/14) at 10 PM Homewok #2 (now on WebAssign) due 9/24 at 10 PM Gauss s Law: Review E = Eid A " = #q in $ 0 Fundamental equation of electostatics (equivalent to Coulomb s Law) Can use it to obtain E fo highly symmetic chage distibutions. Method: evaluate flux ove caefully chosen Gaussian suface : spheical cylindical plana (point chg, unifom sphee, spheical shell, ) (infinite unifom line of chage o cylinde ) (infinite unifom sheet of chage, ) 1

2 Gauss s Law: Examples 1. Spheical symmety (last lectue). 2. Cylindical symmety. Example: infinite unifom line of chage. z Symmety: E indep of z,, in adial diection Gaussian suface: cylinde of length L Eid A = E()2L = q in "# = $L " 0 " 0 E() == 2"# 0 ˆ 3. Plana symmety. y x Example: infinite unifom sheet of chage. z Symmety: E indep of x,y, in z diection Gaussian suface: pillbox, aea of faces=a Eid A = 2EA = q in " = # A E = ẑ 0 0 2" 0 Execise: ty fo E field just outside of a conducto Electic Potential Enegy and Electic Potential Review: Consevation of Enegy (paticle) Kinetic Enegy (K) K = 1 2 mv2 Potential Enegy U: consevative foces (wok independent of path) U(x, y,z) If only consevative foces pesent in system, consevation of mechanical enegy: K + U = constant Examples of consevative foces: Spings: elastic potential enegy U = k sping x 2 2 Gavity: gavitational potential enegy Electostatic: electic potential enegy (today) Examples of nonconsevative foces Fiction, viscous damping (teminal velocity) 2

3 Electic Potential Enegy (I) Compae with gavitational foce (Ch. 13): m1m 2 F 12 = G ˆ 2 12 W = Fid Gm s = 1 m 2 " Gm m 1 2 path f i Gavitational Potential enegy: Gm1m = U 2 Electic Foce: F q q = ke ˆ 2 12 Electic Potential Enegy path independent k q1q e 2 U = Electic Potential Enegy (II) Given two positive chages q and q 0 : q q 0 Initially chages vey fa apat: U i = 0 (we ae fee to define the potential enegy zeo somewhee) To push paticles togethe equies wok (they want to epel). q q 0 Final potential enegy will incease U = U f " U i = W Now, suppose q is fixed at the oigin. What is wok equied to move q 0 fom infinity to a distance away fom q? W = # F us id s = $ # F e id s = $ k eqq # 0 d % = % 2 " " Note: if q negative, final potential enegy negative Paticles will move to minimize thei final potential enegy q q 0 " k e qq 0 3

4 Electic Potential Enegy: Summay Electic potential enegy between two point chages: U is a scala quantity, can be + o - convenient choice: U=0 at = SI unit: Joule (J) U() = k eq 0 q Electic potential enegy fo system of multiple chages: sum ove pais: k e q i q j U() = i< j j ij q 0 q Integal if continuous distibution Example: Thee Chage system What is wok equied to assemble the thee chage system as shown? (q 1 =q 2 =q 3 =Q) Answe: k e 3Q 2 /a (see boad) q 3 a a What if q 1 =q 2 =Q but q 3 =-Q? Answe: -k e Q 2 /a q 1 a q 2 4

5 Electic Potential Enegy: Chage In An Electic Field Chage q 0 is subject to Coulomb foce in electic field E: Wok done by electic foce: W = i f F = q 0 E Fid s = q 0 f Eid s = " #U i U = U f " U i = "q 0 Eid s # i f independent of q 0 Electic Potential Diffeence Electic Potential Enegy: q 0 In a Geneic E. Field U = U B " U A = "q 0 # Eid s = q 0 V B A system potential enegy test souce Electic Potential Diffeence V " U B = # Eid s q A 0 $ = V B # V A 5

6 Popeties of the Electic Potential Results fom consevative natue of the electic foce associated with souce field only (indep. of test chage) units: J/C Volt (V) often called potential, but meaningful only as potential diffeence V B -V A. A convenient point (, eath..) typically chosen as gound ΔV=V-(V A =0) =V scala quantity (no vecto opeations necessay) elated to electic potential enegy by ΔU=q 0 ΔV Execise 2: E. Potential and Point Chages In the configuation shown, find V B -V A B Answe: B V B -V A = k e (q/ B -q/ A ) A A q (See boad) 6

7 Execise 1: Unifom E. Field In the unifom electic field shown: 1. Find potential at B,C,D,G 2. If a chage +q is placed at B, what is the potential enegy U B? 3.If now a q is at B, what is U B? D A V A =0 G C B 4. If a -q is initially at est at G, will it move to A o B? d 5. What is the kinetic enegy when it eaches A? A Pictue to Remembe -q +q E highe V lowe V Field lines always point towads lowe electic potential In an electic field: positive chages ae always subject to a foce in the diection of field lines, towads lowe V negative chage is always subject a foce in the opposite diection of field lines, towads highe V 7

8 Obtaining the Electic Field Fom the Electic Potential Thee ways to calculate the electic field Coulomb s Law E=ΣE i Gauss s Law Deive fom electic potential Fomalism B V = " Eid s # A dv = Eid s = E x dx E y dy E z dz E x = "V "x, E y = "V "y,e z = "V "z o E = #V 8

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