MISN-0-133 GAUSS S LAW APPLIED TO CYLINDRICAL AND PLANAR CHARGE DISTRIBUTIONS GAUSS S LAW APPLIED TO CYLINDRICAL AND PLANAR CHARGE DISTRIBUTIONS by Pete Signell, Michigan State Univesity 1. Intoduction.............................................. 1 a. Oveview................................................ 1 b. Usefulness...............................................1 2. Cylindical Symmety: Line Chage....................1 a. Appoximating a Real Line by an Infinite One........... 1 b. The Gaussian Suface................................... 2 c. The Electic Field....................................... 2 A P ` E 3. Othe Cylindical Distibutions......................... 3 a. Electic Field of a Cylindical Suface................... 3 b. Linea vs. Suface Chage Density.......................4 c. The Coaxial Cable.......................................4 d. Electic Field of the Coaxial Cable...................... 5 4. A Single Sheet of Chage................................ 6 a. Appoximation: An Infinite Sheet....................... 6 b. The Gaussian Suface................................... 6 c. The Electic Field....................................... 7 5. Two Paallel Sheets of Chage.......................... 7 a. Unequal Suface Chage Densities....................... 7 b. Equal Suface Chage Densities..........................8 Acknowledgments............................................8 Glossay...................................................... 8 Poject PHYSNETPhysics Bldg. Michigan State UnivesityEast Lansing, MI 1
ID Sheet: MISN-0-133 Title: Gauss s Law Applied to Cylindical and Plana Chage Distibutions Autho: P. Signell, Dept. of Physics, Mich. State Univ Vesion: 2/28/2000 Evaluation: Stage 0 Length: 1 h; 24 pages Input Skills: 1. Vocabulay: cylindical symmety, plana symmety (MISN-0-153); Gaussian suface, volume chage density (MISN-0-132). 2. State Gauss s law and apply it in cases of spheical symmety (MISN-0-132). Output Skills (Knowledge): K1. Vocabulay: coaxial cable, cylinde of chage, line of chage, sheet of chage, linea chage density. K2. Justify the Gaussian-Suface shapes that ae appopiate fo cylindical and plana chage distibutions. K3. State Gauss s Law in equation fom and define each symbol. Fo cylindical and plana chage distibutions, define needed paametes and then, justifying each step as you go, solve Gauss s Law fo the symbolic electic field at a space-point. Output Skills (Poblem Solving): S1. Given a specific chage distibution with cylindical o plana symmety, use Gauss s law to detemine the electic field poduced by the chage distibution. Post-Options: 1. Electic Fields and Potentials Acoss Chage Layes and In Capacitos (MISN-0-134). 2. Electostatic Capacitance (MISN-0-135). THIS IS A DEVELOPMENTAL-STAGE PUBLICATION OF PROJECT PHYSNET The goal of ou poject is to assist a netwok of educatos and scientists in tansfeing physics fom one peson to anothe. We suppot manuscipt pocessing and distibution, along with communication and infomation systems. We also wok with employes to identify basic scientific skills as well as physics topics that ae needed in science and technology. A numbe of ou publications ae aimed at assisting uses in acquiing such skills. Ou publications ae designed: (i) to be updated quickly in esponse to field tests and new scientific developments; (ii) to be used in both classoom and pofessional settings; (iii) to show the peequisite dependencies existing among the vaious chunks of physics knowledge and skill, as a guide both to mental oganization and to use of the mateials; and (iv) to be adapted quickly to specific use needs anging fom single-skill instuction to complete custom textbooks. New authos, eviewes and field testes ae welcome. PROJECT STAFF Andew Schnepp Eugene Kales Pete Signell Webmaste Gaphics Poject Diecto ADVISORY COMMITTEE D. Alan Bomley Yale Univesity E. Leonad Jossem The Ohio State Univesity A. A. Stassenbug S. U. N. Y., Stony Book Views expessed in a module ae those of the module autho(s) and ae not necessaily those of othe poject paticipants. c 2001, Pete Signell fo Poject PHYSNET, Physics-Astonomy Bldg., Mich. State Univ., E. Lansing, MI 48824; (517) 355-3784. Fo ou libeal use policies see: http://www.physnet.og/home/modules/license.html. 3 4
MISN-0-133 1 GAUSS S LAW APPLIED TO CYLINDRICAL AND PLANAR CHARGE DISTRIBUTIONS by Pete Signell, Michigan State Univesity 1. Intoduction 1a. Oveview. In this module Gauss s law is used to find the electic field in the neighbohood of chage distibutions that have cylindical and plana symmety. Fo each of these two symmeties, useful Gaussian sufaces ae easily constucted. Once an appopiate Gaussian suface is constucted, the electic field is easily found fom Gauss s law: 1 E ˆn ds = 4πk e q S (1) whee q S is the net chage enclosed by the Gaussian suface S and k e is the electostatic foce constant. 1b. Usefulness. It is vey useful to know the electic field in the neighbohood of cylindical and plana chage distibutions, fo these geometies ae the ones used in coaxial cables and capacitos. Knowing the electic fields helps one detemine how these devices will eact in electonic cicuits. In addition, the same geneal ideas ae used in detemining the magnetic fields poduced in solenoids, tansfomes, coaxial cables, chokes, and tansmission lines. 2. Cylindical Symmety: Line Chage 2a. Appoximating a Real Line by an Infinite One. When dealing with a line of chage, we will teat it as though its ends had been extended to infinity. This appoximation makes the esulting electic field especially simple and easy to solve fo. The solutions we get fo the infinitely long line will be applicable to the finite-line case fo electic field points that ae much close to the middle pat of the line than to its ends. Fo pactical applications the infinite line is almost always a good appoximation to the actual finite line. 1 See Gauss s Law and Spheically Symmetic Chage Distibutions (MISN-0-132) fo an intoduction to Gauss s law and the ules fo using it. MISN-0-133 2 Gaussian suface line of chage Figue 1. A cylindical Gaussian suface is used to apply Gauss s law to a line of chage. 2b. The Gaussian Suface. Fo an infinitely long line with unifom linea chage density 2 along it, the pefeed Gaussian suface is cylindical (see Fig. 1). This follows fom taking the two ules fo constucting Gaussian sufaces and combining them with knowledge of the electic field s diections and equi-magnitude sufaces. 3 The axis of the cylindical suface is along the line of chage, while the suface s adius is that of the point at which you wish to know the electic field. The length of the cylindical suface is immateial. 2c. The Electic Field. Applying Gauss s law, Eq. (1), to the case of a staight line of chage with unifom linea chage density (chage pe unit length) λ, we will show that the magnitude of the electic field at a distance fom the line is: E = 2k e λ. (2) Poof: If the length of the cylindical Gaussian suface is L, then the chage enclosed by the suface is: q S = λ L. (3) The component of the electic field nomal to eithe flat end of the closed cylindical suface is zeo, but the component nomal to the cylindical 2 The tem linea chage density means the chage is being descibed as a cetain amount of chage pe unit length along the wie. This is in contast to volume chage density whee the chage is descibed as a cetain amount of chage pe unit volume within the wie. Fo unifom coss-sectional distibutions, the linea chage density equals the coss-sectional aea times the volume chage density. 3 Fo the two ules fo constucting Gaussian sufaces, see Ref. 1. Fo deivation of the electic field diections and equi-magnitude sufaces see Electic Fields fom Symmetic Chage Distibutions (MISN-0-153). 5 6
MISN-0-133 3 pat of the suface is just the field itself: E ˆn ds = E ˆn ds E ˆn ds = E cyl. ends cyl. ds 0 = E(2πL). (4) Using Gauss s law, Eq. (1), to combine Eqs. (3) and (4), we obtain the solution, Eq. (2). Of couse in a eal poblem ou solution would be valid only in the egion whee the distance to the line of chage is much smalle than the distance to the line s neaest end. As an amusing aside, notice that Eq. (2) says that the sound of a long line of taffic will only die off as 1 athe than the 2 one obtains fo a point souce. 3. Othe Cylindical Distibutions 3a. Electic Field of a Cylindical Suface. A cylindical suface with finite adius, constant suface chage density, and infinite extent, has an electic field whose pefeed Gaussian sufaces ae identical to those fo an infinite chaged line (see Fig. 2). This is because both the line and the cylindical suface have the same geometical symmety and hence the same electic field diections and equi-magnitude sufaces. 4 Fo a chaged suface of adius R and suface chage density σ, the amount of chage 4 See Electic Field fom Symmetic Chage Distibutions, (MISN-0-153), the section on infinitely long cylindical chage distibutions. R Gaussian suface cylindical suface of chage Figue 2. The Gaussian suface fo a cylindical suface chage distibution (on a cylindical suface of adius R). MISN-0-133 4 enclosed by a Gaussian suface of adius and length L is: q S = 2πR L σ fo > R = 0 fo < R Exactly as in the case of the line of chage, the integal of the nomal component of the electic field ove the Gaussian suface is: E ˆn ds = (2πL)E. (6) Then using Eqs. (5) and (6) in Gauss s law, Eq. (1), we find: σr E = 4πk e fo > R = 0 fo < R 3b. Linea vs. Suface Chage Density. We may descibe the chage distibution on a cylindical suface as eithe a suface chage density o a linea chage density. The suface chage density σ is the chage pe unit aea on the cylindical suface: σ = q (8) 2πl whee 2πl is the suface aea of a cylinde of adius and length l. The linea chage density λ is the (total) chage pe unit length along the cylindical suface: λ = q = 2πσ. (9) l Show that if q = 1.0 10 6 C, = 1.0 cm, and l = 1.0 m, then σ = 1.6 10 5 C/m 2 and λ = 1.0 10 6 C/m. 3c. The Coaxial Cable. A coaxial cable, such as that used to tanspot TV signals o the signal fom a pickup to a steeo amplifie, consists of two metallic conductos with cylindical symmety, shaing a common cylinde axis and sepaated by some kind of insulato. The cente cylinde is usually a solid coppe wie while the oute one is usually a sheath of baided wie (see Fig. 3). This constuction makes the cable mechanically flexible. As fa as the electical popeties ae concened, thee might as well be two concentic cylindical sufaces 5 as shown in Fig. 4. The magnitude of the linea chage density on the two cylindes is the same, so the 5 Electostatic chages eside on the sufaces of metallic conductos: see Electostatic Popeties of Conductos (MISN-0-136). (5) (7) 7 8
MISN-0-133 5 MISN-0-133 6 dielectic (usually white, flexible) cente wie (solid) baided wie sheath plastic skin Figue 3. A coss-sectional view of a typical coaxial cable. I II III Figue 4. A cosssectional view of the chage sufaces in a coaxial cable. The adius of the inne cylinde is exaggeated fo the pupose of illustation. magnitude of the suface chage density is highe on the inne cylinde. Fo odinay uses the chages on the two cylindes ae of opposite sign. 3d. Electic Field of the Coaxial Cable. Gauss s law shows that the electic field of a chaged coaxial cable is zeo except between the conducting cylindical sufaces, whee it is equal to the field poduced by the inne cylinde. Applying Gauss s law to the coaxial cable s vaious egions, as shown in Fig. 4, the electic field is eadily found to be: E I = 0 λ E II = 2k e ˆ E III = 0 Help: [S-1] (10) In egion II of Fig. 4, λ is negative, so that paticula electic field is diected adially inwad. a) b) ` E n^ n^ C A B n^ P ` E Figue 5. (a) A coss-sectional view of a Gaussian suface (dashed lines) fo an infinite plane of chage; and (b) a theedimensional view of the Gaussian suface. 4. A Single Sheet of Chage 4a. Appoximation: An Infinite Sheet. We will estict ouselves to the case of a unifom plana chage distibution of infinite extent: in othe wods, a flat sheet with a unifom suface chage density that extends to infinity. These estictions make the esulting electic field especially simple and easy to detemine. The solutions we get will be valid fo any application in which the sheet of chage can be appoximated by an infinite sheet with the same suface chage density. This appoximation will be a good one when the distances fom elevant electic field points to the edges of the physical sheet ae all much lage than the distance to the neaest point on the sheet of chage. Thus the edges will look an almost infinite distance away (in compaison). This will be the case fo impotant chage-stoing components in electonic cicuits. 4b. The Gaussian Suface. Fo a unifom plana chage distibution of infinite extent, all pats of the pefeed Gaussian suface can be shown to be eithe paallel o pependicula to the plane of the chage. This equiement would be satisfied, fo example, by a box-like suface that is cut by the plane into two equal boxes (see Fig. 5). The paallel o pependicula equiement follows fom taking the two ules fo constucting Gaussian sufaces and combining them with ou knowledge of the electic field s diections and equi-magnitude sufaces. Any suface that satisfies the paallel o pependicula equiement is acceptable, C A B P ` E 9 10
MISN-0-133 7 but ectangula and cylindical boxes ae the easiest to use in computing the suface aeas and volumes that ente into Gauss s law. 4c. The Electic Field. Applying Gauss s law to a flat infinite sheet with unifom suface chage density σ, we find that the magnitude of the electic field is eveywhee the same: E = 2πk e σ. (11) The diection of the field is nomal to the sheet of chage, diected away fom the sheet fo a positive chage density and towad the sheet fo a negative chage density. If the aea of one end of the box-like Gaussian suface is A, then the chage enclosed by the suface is: q S = σ A. (12) The component of the electic field nomal to the side of the Gaussian suface is zeo on the fou sides that cut though the plane and equal to the electic field on the othe two sides: E ˆn ds = 2 E A. (13) Using Gauss s law to combine Eqs. (12) and (13) we obtain Eq. (11), the solution. Of couse in a eal poblem the constancy of the electic field is esticted to egions whee the distance to the sheet of chage is much smalle than the distance to the sheet s neaest edge. 5. Two Paallel Sheets of Chage 5a. Unequal Suface Chage Densities. Gauss s law can be easily applied to the case of two infinite paallel sheets having unifom suface chage densities σ and σ, espectively. To obtain the electic field at some paticula point, apply Gauss s law to each of the sheets sepaately and then add the fields fom the two sheets vectoially. Note that the two Gaussian sufaces have one side in common. You should obtain the answes: Help: [S-3] E = 2πk e (σ σ ), (outside the planes). (14) E = 2πk e (σ σ ), (between the planes). (15) Note that these equations become paticulaly simple when σ and σ ae equal. Note also that this poblem has no symmety plane so a single pefeed Gaussian suface could not be dawn: the two ules fo constucting such a suface could not be satisfied with only ou usual pio knowledge of the field. MISN-0-133 8 5b. Equal Suface Chage Densities. Fo two infinite paallel sheets of chage with identical suface chage densities, σ, we can apply Gauss s law using a single Gaussian suface. Thee is a plane of symmety that is paallel to the two sheets and half way between them, so the Gaussian suface must be symmetical with espect to that plane of symmety. In pactical tems, the symmety plane must cut the boxlike suface into two identical box-like sufaces. Note how this symmety of the Gaussian suface with espect to the poblem s symmety plane ensues that the two ules fo constucting the pefeed suface can be satisfied. Help: [S-2] If one end of the suface has aea A, the chage enclosed by the suface is: q S = 2 σ A. (16) Then Gauss s law poduces: and E = 4πk e σ, (outside the sheets), (17) E = 0, (between the sheets). These two equations agee with Eqs. (14) and (15). Acknowledgments I would like to thank Pofesso J. Linnemann fo a valuable suggestion. Pepaation of this module was suppoted in pat by the National Science Foundation, Division of Science Education Development and Reseach, though Gant #SED 74-20088 to Michigan State Univesity. Glossay coaxial cable: an electical cable consisting of two metallic concentic cylindes sepaated by some kind of insulato. The electic field is zeo both inside the inne cylinde and outside the oute cylinde. cylinde of chage: a chage distibution with cylindical symmety. Fo a cylinde of chage with constant density extending to infinity, the associated electic field falls off as (1/) outside the suface ( is the adius fom the axis of the cylinde). line of chage: a chage distibution along a staight line. This is a special case of cylinde of chage. 11 12
MISN-0-133 9 linea chage density: the chage pe unit length along a line. sheet of chage: a chage distibution in a plane. Fo a plane of chage with constant density extending to infinity in all diections, the associated electic field is eveywhee constant and nomal to the plane. MISN-0-133 PROBLEM SUPPLEMENT Note: Poblems 7 and 8 also occu in this module s Model Exam. PS-1 1. Two paallel lines of chage, shown coming out of the page in the sketch, ae a distance 2.0d apat. If both ae positively chaged, find the electic field as a function of y at points along the pependicula bisecto of the line connecting the two [find E(y) fo x = 0]. y d d x 2. An infinitely long cylinde of chage has a adius R and a volume chage density ρ. Find the electic field in the egions < R and > R and show that both lead to the same esult at = R. Help: [S-9] 3. Find the electic field inside and outside a hollow cylinde of chage whose adius is R, and whose linea chage density is λ. R R 13 14
MISN-0-133 PS-2 MISN-0-133 PS-3 4. A cylinde of chage has volume chage density ρ and adius R 1. Outside it and concentic with it is a cylindical suface of chage with adius R 2. The chage on this oute suface has a linea chage density λ. Find the electic field in each of the thee egions defined by the cylinde sufaces at R 1 and R 2. Show that the esult in Poblem 3 can be obtained fom the esults in this Poblem by letting R 1 = 0, R 2 = R. 5. Find the electic field in the fou egions defined by the thee infinite planes (sheets) of chage in the sketch. y^ I II III IV x^ s - s s R 1 R 2 7. Use Gauss s law to find the electic field outside and inside two lage paallel plates with equal suface chage densities σ on the plates and with volume chage density ρ between the plates. The distance between the plates is d. Help: [S-11] 8. Use Gauss s law to find the electic field in each of the thee egions defined by two coaxial cylindical sufaces, each with linea chage density λ, and with a unifom volume chage density ρ inside the inne cylindical suface. The adii of the two cylindical sufaces ae R 1 and R 2 (see diagam below). R 2 R 1 volume chage x^ 6. An infinite slab of chage of thickness d has a unifom chage density ρ, as shown below in coss section in the sketch. Use Gauss s law to find the electic field inside and outside the slab. Help: [S-4] y d x 15 16
MISN-0-133 PS-4 MISN-0-133 PS-5 Bief Answes: E I = 2πk e σˆx ; Help: [S-8] 1. E x = 0 λy E y = 4k e (d 2 y 2 ) Help: [S-10] 2. E() = 2πke ρ ˆ fo < R Help: [S-6] ρr E() 2 = 2πk e ˆ fo > R At the suface: E(R) = 2πke ρr ˆ fo = R 3. Region 1 ( < R 1 ): E() = 2πke ρ ˆ Help: [S-7] Region 2 (R 1 < < R 2 ): E() = 2πke ρr 2 1 Region 3 (R 2 < ): E() ρπr = 1 2 λ 2ke ˆ ( ) ρr 2 1 = 0 ; E = 0 (fo < R) ) 4. lim R 1 0 2πk e lim R 1 0 ( ρr1 2 λ 2πk e 2k e ˆ λ = 2k e ; E() λ = 2k e ˆ (fo > R) 5. In the figue below, note that each ow of fou aows shows the field that would be poduced by just one plane of chage alone (see the annotations down the ight side of the figue). E II = 2πk e σˆx ; EIII = 2πk e σˆx ; EIV = 2πk e σˆx. 6. Inside : E = 4πke ρy ŷ ; Outside : E = 2πke ρd ŷ 7. Outside: Eight = 2πk e (2σ dρ) ˆx E left = 2πk e (2σ dρ) ˆx Inside: E = 4πke ρ x ˆx, whee x is measued fom the symmety plane between the plates. 8. Inside egion: E() = 2πke ρ ˆ Between egion: E() = 2ke πr 2 1ρ λ Outside egion: E() = 2ke πr 2 1ρ 2λ ˆ ˆ I II III IV 1 2 3 field due to 1 field due to 2 field due to 3 17 18
MISN-0-133 AS-1 MISN-0-133 AS-2 SPECIAL ASSISTANCE SUPPLEMENT S-3 (fom TX-5a) Multiply all shown values by 2πk e : S-1 (fom TX-3d) S I EI ˆn ds = 4πk e q SI = 0 E I ˆn = 0 on cyl. suf. E I = 0 S II EII ˆn ds = 4πk e q SII = 4πk e λl E II ˆn(2πl) = 4πk e λl E II = 2k e λ ˆ S III EIII ˆn ds = 4πk e q SIII = 0 E III ˆn = 0 on cyl. suf. E III = 0 S-2 (fom TX-5b) S E ˆn ds = 4πk e q S = 4πk e 2σA on ends: E ˆn(2A) = 4πke 2σA outside the planes: E = 4πke σ ˆn S 1 E ˆn ds = 4πke q S = 0 on ends: E ˆn(2A) = 0 between the planes: E = 0 Region I Region II Region III n^ Cylindical Gaussian Suface n^ ' ' ' ' S-4 (fom PS-Poblem 6) ' - ' ' Since the slab has plana symmety, the field diection is eveywhee nomal to the slab and the equi-magnitude sufaces ae paallel to the slab faces (see MISN-0-153). When you constuct Gaussian sufaces, take advantage of the eflection symmety about the x-z plane by choosing the two faces that ae paallel to the slab to be equidistant fom the slab as well. That way you can wite the suface integal of E as: E ds = 2 E A S since the field is unifom ove the two Gaussian suface aeas, and the field must be the same on each suface by symmety. S-5 (fom [S-10]) Daw a sketch of the situation and on it mak the given quantities. Then figue out and mak on the sketch the θ and we use hee: E y (total) = E y (fom #1) E y (fom #2) E y (fom #1) = E(fom #1) cos θ E y (fom #2) = E(fom #2) cos θ E(fom #1) = E(fom #2) = 2k e λ/() whee = (d 2 y 2 ) 1/2, λ is the chage pe unit length along each wie, and cos θ = y/. E y (total) = 4k e λy/( 2 ) = 4k e λy/(d 2 y 2 ) 19 20
MISN-0-133 AS-3 MISN-0-133 ME-1 S-6 (fom PS-Poblem 2) Chage volume density is chage pe unit volume and in the MKS system is measued in C/m 3. Then: total chage = chage volume density volume, whee all thee quantities efe to inside the suface. S-7 (fom PS-Poblem 4) Completely solve Poblems 1 and 2 fist. Linea chage density is chage pe unit length and in the MKS system is measued in C/m. It is the amount of chage pe unit length down the entie cylindical suface. S-8 (fom PS-Poblem 5) Use Gauss s Law sepaately on each plane (as though the othe did not exist) to get: EI (due to #1), EI (due to #2), and E I (due to #3). Then add them to get E in egion I, hee labeled E I. S-9 (fom PS-poblem 2) The concept that has caused students touble in the past (in Poblem 2) is diectly and clealy handled in this module s text. Read and undestand it thee. S-10 (fom PS-poblem 1) Fist, ty you vey best to solve this poblem without Special Assistance. Go back and wok though the text again, this time paying special attention to sections elevant to this poblem. Remembe that you will not have the Special Assistance available at exam time so you need to lean how to wok without it. If you ty and ty and tuly fail, then ty the Special Assistance in [S-5]. S-11 (fom PS-poblem 7) Calculating the enclosed chage involves techniques you used in the two pevious poblems. MODEL EXAM 1. See Output Skill K1 in this module s ID Sheet. 2. Use Gauss s law to find the electic field outside and inside two lage paallel plates with equal suface chage densities σ on the plates and with volume chage density ρ between the plates. The distance between the plates is d. 3. Use Gauss s law to find the electic field in each of the thee egions defined by two coaxial cylindical sufaces, each with linea chage density λ, and with a unifom volume chage density ρ inside the inne cylindical suface. The adii of the two cylindical sufaces ae R 1 and R 2 (see diagam below). R 2 Bief Answes: R 1 1. See this module s text. volume chage 2. See this module s Poblem Supplement, Poblem 7. x^ 3. See this module s Poblem Supplement, Poblem 8. 21 22
23 24