Aaa Hal ARC 103 Haq Mou Hill 114 Mug Seh PHY LH Sen Zzz SEC 111

Transcription

1 The Fist Midtem!!! The fist common hou midtem exam will be held on Thusday Octobe 5, 9:50 to 11:10 PM (at night) on the Busch campus. You should go to the oom coesponding to the fist 3 lettes of you last name. If you have a conflict with the exam time, please contact Pof. Montalvo with you entie schedule fo the week of Octobe 2 at you ealiest convenience but not late than 5:00 pm on Wednesday, Septembe 27. Aaa Hal ARC 103 Haq Mou Hill 114 Mug Seh PHY LH Sen Zzz SEC 111

2 Gaduate Student Physics Tutos The physics gaduate students have indicated an inteest in tutoing students in undegaduate physics couses. If you would be inteested in having a tuto, please feel fee to contact them. The tutoing is fo a fee that you will need to negotiate with the tuto. Gaduate Student Tuto List on the couse website. Thee is in addition fee pee tutoing offeed at the Rutges Leaning Centes 2

3 Lectue 5. Electostatic Potential Enegy Coulomb s Law and Supeposition Pinciple sufficient to solve any electostatic poblem. Gauss Law simplifies calculation of electic fields fo symmetic chage distibutions. Why do we need anything else? - Electostatic foces ae consevative, and enegy consevation is a vey poweful tool fo solving a wide ange of poblems. - The concept of the potential enegy of the electostatic field is the fist step towads the appeciation of the enegy of electomagnetic field. 6

4 Recall: Potential Enegy in Gavitational Field F g F a uu b Wok done by us while lifting the body fom a to b at a constant speed: U W "uuu b a b = F uu dl a dl = dd Wok done by the gav. foce F g : W "ffffff b a b = F g dl a Change in potential enegy is equal to the wok done by extenal foces ( us ) to move an object in a gavitational field (assuming that the kinetic enegy emains constant, F g = F uu ). U b U a W a b "uuu = F uu dl a b b = F g dl a Units: Joules, J Potential enegy: always the enegy of inteaction between masses, not a chaacteistic of a single mass (the potential enegy of the system Eath + mass m ). 7

5 Gavitational Potential Enegy of the system Eath + astonaut F g F a uu b Potential enegy with espect to (we dag the astonaut fom to ): U U = F uu dl = G m Em dd 2 = G m Em dd 2 = G m Em U 2 1 U + K K 2 K 1 = 0 U + K = ccccc Enegy consevation 8

6 Refeence Point(s) U = F uu dl sooo.ppppp Refeence point: matte of convenience, only U mattes (because only foces can be measued, and the foces depend on U, not U). F = U The sea level ( = R E ) as a efeence: U U R E = Gm E m 1 1 R E Gm E m R E R E 2 = m Gm E R E 2 h = mm h R E = h R E g ( ) U h = mg h 9

7 Consevative Vecto Fields Gavitational and Electostatic Fields ae consevative. The eason: both fields ae cental (a cental foce depends only on the distance between inteacting objects and is diected along the line joining them). In cental vecto fields, U depends only on the initial and final points of the tajectoy: F E dl = 0 - fo any loop No closed E field lines in electostatics: 10

8 Wok in Electostatic Field Conside two point chages, +q 1 and +q 2. Place +q 1 at the oigin. q 1 q 2 Let s bing q 2 fom to ( ) P at a distance fom the oigin. The wok done by us: W "uuu = F uu dl U 1/ along the geen tajectoy F uu dl = k q 1q 2 d 2 U = 0 - ou ef. point q 2 E 1 F uu q 2 E 1 = kq 1 q = k q 1q 2 We woked against the electostatic foce of epulsion and inceased the potential enegy of this chage distibution. 11

9 Electostatic Potential Enegy U = F uu dl = qe dl sooo.ppppp sooo.ppppp - the potential enegy of the system chage q + extenal electostatic field E ( )P q 1 q 2 ( )Q ( )P ( )Q U = q 2 E 1 dl = q 1 E 2 dl = k q 1q 2 U E ttt = K + U U ( )P ( )Q ++ o o -+ 12

10 Example U = eee QE dl sssss 1,0 1, 2 = Q E x, y dl + E x, y dl 0,0 1, = Q 2x + 3y dd x + 2x + 3y dd y = Q 2dd + 3dy = J = 12 J 14

11 Fom U to F U = F dl ee.ppppp One-dimensional case: F x = du x dx x ( )P +q q ( )Q U = k q 1q 2 = k q2 U 0 F = du dd (spheical symmety) du /dd positive, F is diected along. (as it should be, because of attaction) Q: is it possible to find E() if we know U() only at this paticula point? 15

12 Choice of Refeence Point The efeence point can be any point. By shifting the efeence point, we add the same constant values to the potential enegies at all othe points: U = qe dl U This ambiguity shouldn t bothe us, because only the potential enegy diffeence is meaningful, and the diffeence doesn t depend on the efeence point. Localized ( finite ) chage distibution. The natual choice is to set U=0 at an infinitely distant point. Again, it s just a matte of convenience. Fo many poblems, it s convenient to set the potential enegy of chages on the gound to zeo. O 17

13 Infinite Chage Distibutions - chage distibution that extends to infinity (e.g., a chaged infinite plane). The choice of an infinitely distant efeence point is inconvenient all electostatic enegies would be infinitely lage. The emedy is to choose less emote efeence point. Example: unifomly chaged plane. y 0, 0 Convenient efeence point: any point within the plane (x=0). All points within the plane have the same U (we can move the chage ove the plane without doing any wok). x 0, y x, y U = U x > 0, y U 0,0 x,y = qe x dl U(x) 0,0 x y = qqx y dy 0 x qqx x dx 0 = q σ 2ε 0 x = 0 18

14 q Supeposition: seveal inteacting chages If the test chage inteacts with seveal chages, the potential enegy of the inteaction between the test chage and all othe chages is an algebaic sum of U 0i : q 1 q 3 03 q 0 U 0 = k q 0q q 0q q 0q 3 03 = q 0 k q i 0i i Example: a positive test chage inteacting with a dipole U 0 = k q 0q k q 0q 2 02 = kq 0 q q q 0 q U 0 U 0 x x 19

15 Conclusion Consevative vecto fields Potential enegy. Electostatic potential enegy. Calculation of U fo simple chage distibutions. Next time: Lectue 6. Calculation of Electic Potentials

16 Appendix I. Consevative Vecto Fields Consevative vecto fields: the wok by the field on a chage depends only on the initial and final points of a tajectoy, but not on the shape of the tajectoy. Math involved: cccc a = 0 a dl = 0 - ciculation of a vecto field cccc E = E z E y x + E x z E z x y + E y x E x y z - looks scay But we can tell at a glance if the cul is zeo o not: think of a vecto field as a wate flow and place a paddle wheel at the point in question. If the wheel otates, the cul is non-zeo. y y a x, y = yy a x, y = xy x E y = E y = 0 x E y = 1 cccc a = 0 cccc a 0 21

17 Appendix II: Total enegy of inteactions between seveal chages q 2 02 The total wok equied to assemble this chage distibution: 12 q 1 13 q 3 03 q 0 U Σ = q 0q 1 4πε q 0q 2 4πε q 1q 2 4πε = 1 4πε 0 q iq j ii i<j - all possible pais, but each pai we count just once. q a q a q q Example: fou equal chages in the cones of a squae. U e ttttt = q iq j 4πε 0 ii all ppppp each pai counts only once = 4 q2 4πε 0 a + 2 q 2 4πε 0 a 2 The enegy is positive (the chages epel each othe), the system can do some wok if we let the chages go. On the othe hand, the potential enegy of one of these chages in the field due to the othe thee chages: U e = 2 q2 4πε 0 a + q 2 4πε 0 a 2 22