Brown University PHYS 0060 ELECTRIC POTENTIAL

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Ethelbert Richard
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1 INTRODUCTION ELECTRIC POTENTIL You have no doubt noticed that TV sets, light bulbs, and othe electic appliances opeate on 115 V, but electic ovens and clothes dyes usually need 220 V. atteies may be ated at a hamless 1.5, 6, 9, o 12 V, but a high-tension electic tansmission line may povide electic powe at V. Now just what physical quantity is measued by all these volts? How do volts elate to foce, enegy, and powe, about which you have leaned in ealie modules? The answe is that volts measue electic potential diffeence (sometimes called voltage ), which is deived fom the potential enegy acquied by electically chaged objects as a esult of the electic foces they expeience. Even though you familiaity with volts pobably stems fom electic powe supplied to you household, you intoduction to the concept of electic potential in this module will be in the context of the inteaction of stationay (static) electic chages. PREREQUISITES efoe you begin this module, you should be able to: *Calculate wok and apply the wok-enegy theoem in solving poblems (needed fo Objective 1 of this module) *Calculate potential enegy and identify consevative foces (needed fo Objectives 1 & 2 of this module) *Calculate electic foce (needed fo Objective 1 of this module) *Intepet a line integal and find the deivative of a function of one vaiable (needed fo Objectives 2 & 4 of this module) *Find the electic field using Gauss Law and descibe the electic field nea conductos (needed fo Objectives 3 & 5 of this module) *Find the electic field using Coulomb s law and the supeposition pinciple, and use field lines to descibe the electic field (needed fo Objectives 3 & 5 of this module) Location of Peequisite Content Wok and Enegy Module Consevation of Enegy Module Coulomb s Law and the Electic Field Module Calculus Review Flux and Gauss Law Module Coulomb s Law and the Electic Field Module Flux and Gauss Law Module

2 LERNING OJECTIVES fte you have masteed the content of this module, you will be able to: 1. Definition Relate electic potential to (a) wok done on a displaced chage, (b) the electic field, and (c) electic potential enegy. Use the electon volt to expess enegy and solve simple poblems applying enegy consevation. 2. Consevative Field State and intepet the consevative natue of the electostatic field. 3. Finding Potentials fom Chages Use the definition and/o the supeposition pinciple fo finding the electic potential caused by (a) one o moe given point chages, and (b) continuous chage distibutions with plana, cylindical, o spheical symmety. 4. Finding Fields fom Potentials Detemine the electic field when given an electic potential that is a function of one position vaiable only. 5. Equipotential Sufaces Use equipotential sufaces and field lines fo descibing the potential and field semi-quantitatively nea seveal given point chages and/o simply shaped metallic sufaces. 2

3 GENERL COMMENTS 1. The Electon Volt The electon volt (ev) is a unit of enegy commonly used in atomic physics. It has the advantage that when an electon moves in an electic field acoss a potential diffeence of, say, 150 V, its kinetic enegy changes by 150eV. In othe wods, the potential diffeence gives the enegy diectly in electon volts, if you ae dealing with a poton, electon, o othe singly chaged atomic paticle. Fo household puposes, the electon volt is impactically small. The new system of Intenational Units (SI) asks that the electon volt only be used in atomic physics. Othewise it should be conveted to joules (1 ev = x J). Remembe to convet to SI units befoe using masses in kilogams and velocities in metes pe second with an enegy oiginally given in electon volts. 2. Potential as a Function of Position cetain point at infinity, at the coodinate oigin, o at some othe point having symmety in elation to the chages and fields is chosen as the efeence point in ode to give potential as a function of position in elation to that point absolute potential ). Then the potential is simply a function of the coodinates of any given point with the efeence point at zeo. 3. Consevative Natue of the Electic Field e electostatic foces consevative? You know that a foce field is consevative (pemits the definition of a potential enegy function) if and only if (a) the wok done by the foce on a paticle moved fom point to point is independent of the path taken fom a to ; o (b) the wok done by the foce on a paticle moving though a closed path back to its stating point is zeo. Since the electostatic foce at any point is diectly popotional to the electic field at that point, F e (d ) = qe (d ) (1) The two equivalent conditions on the wok stated above lead to two equivalent conditions on the field (a) W = F e (d ) dl is independent of path fom to (2) implies that E (d ) dl is independent of path fom to. (3) (b) W = F e (d ) dl = 0 by any closed path (4) 3

4 implies that E (d ) dl = 0 by any closed path (5) The line integal in Eqs. (4) and (5) along a closed path is sometimes indicated by an integal sign with a little cicle: E (d ) dl = 0 (6) (Please do not confuse this symbol with the same symbol used by many texts to epesent a suface integal; ove a closed suface as in Gauss law. You will have to distinguish the two symbols by looking caefully at the infinitesimal element unde the integal sign and seeing whethe it efes to a suface o to a displacement.) We shall now show that the Coulomb field caused by a single point chage is consevative by evaluating the integal in Eq. (3) and obseving that its value depends only on the end points, not on the path. The magnitude of the electic field caused by the point chage q depends only on the distances (not on the entie vecto d ) fom the chage and is E(d) = kq 2 (7) E 3 dd 4 q dl 4 (a) (b) Figue 1 whee k = 1 and is the distance. The diection of field is adial. Figue 1 illustates the 4πε 0 geometical elationships. In Figue 1(a) you can see points and, a paticula path we have chosen, and aows epesenting fou electic field values E and fou infinitesimal line elements dl. We must now calculate E (d ) dl, which we shall do with the help of the enlaged diagam in Figue 1(b). We have indicated that angle θ 4 between the path and the adial diection and also the adial incement dd 4 = dl 4 cos θ 4. Using the definition of the dot poduct, we find 4

5 E (d ) dl = E(d)dl cos θ = E(d)dd = kq 2 dd (8) In othe wods, because the field is adially diected, only the adial incements and not the angula incements of the path contibute to the line integal. (In some texts, the cuved path is appoximated by a stai-step path. They then fail to show that the stai-step integal appoaches the line integal along the cuved path; assuming that this is tue comes close to assuming that the integal is independent of the path.) With the integand in Eq. (8), we can calculate the integal ove the adius vaiable, which vaies fom to : E (d ) dl = E(d)dd = kq 2 dd = kq(d 1 d 1 ) (9) Evidently, this esult depends only on the end points and not on the path. Ou poof is heeby completed. Ou conclusion of path independence can be extended easily to an electic field that is caused by seveal point chages. Since such a field is obtained by adding the fields fom the individual chages, the line integal can be expessed as a sum of line integals like that in Eq. (9), fo each of which path independence has been established. It is not so easy to extend the poof to the fields caused by continuous chage distibutions. s a matte of fact, moe advanced teatments of the theoy of static electic fields always begin with the postulate that the fields ae consevative. PROLEM SET WITH SOLUTIONS (1) 1. Conside a egion of constant electic field in the x diection, with the field having magnitude 400 N/C. The fou points,, C, and D have the coodinates (5.0, 0.00, 0.00) m, (3.00, 2.00, 0.00) m, (9.0, 0.00, 2.00) m, and (3.00, 0.00, 0.00) m. Thee is no gavitational field. a. n extenal agent slowly moves a body with chage 0.35 C fom to. How much wok is done on the body by the electic field, and how much is done on the body by the extenal agent? Use a sketch to illustate. b. nswe the questions in pat (a) fo displacement of the same body (i) fom to C, (ii) fom to D, and (iii) fom to D. c. bead of mass 0.60 kg and chage C is pemitted to slide along a fictionless wie between points and C. t which of the two points should it be eleased fom est in ode that it will get to the othe one, and with what speed will it aive? 5

6 Solution d. Find the diffeences in electic potential between points and, points C and, and points D and ; compae them with you answes to pats (a) and (b). (a) See sketch in Figue 2. Fom the definition, W = F e dl Fo a constant field, as in this case, W = F e dl = qe Δl = (0.35)(400)( 2.00) = 280 J done by field The extenal agent exets foce -F e, hence does +280 J of wok. (b) i. W C = qe Δl C = (0.35)(400)(+4.0) = +560 J; W C(agent) = 560 J ii. W D = qe Δl D = (0.35)(400)( 2.0) = 280 J; W C(agent) = +280 J iii. W D = qe Δl D = (0.35)(400)(0) = 0 J; W C(agent) = 0 J (c) The negative chage in this pat expeiences a foce to the left, hence it must be eleased at C and will acquie speed sliding towad. The kinetic enegy gained is equal to the wok done on it by the electic field, which is W C = qe Δl C = ( 1.20)(400)(6.0) = 2880 J Hence K = 1 2 mv2 = 2880 J, and v = = 9600 = 98 m s y E = 400ı N/C D Δl Figue 2 C x (d) V V = E Δl = 800 V V C V = E Δl C = 1600 V V V D = E Δl D = 0 V. W = q(v V ). W C = q(v C V ). W D = q(v V D ) Electic Field electons 7500 V Figue 3 6

Voltage ( = Electric Potential )

Voltage ( = Electric Potential ) V-1 of 9 Voltage ( = lectic Potential ) An electic chage altes the space aound it. Thoughout the space aound evey chage is a vecto thing called the electic field. Also filling the space aound evey chage