Module 12: Cuent and Resistance 1 Table of Contents 6.1 Electic Cuent... 6-2 6.1.1 Cuent Density... 6-2 6.2 Ohm s Law... 6-4 6.3 Electical Enegy and Powe... 6-7 6.4 Summay... 6-8 6.5 Solved Poblems... 6-9 6.5.1 Resistivity of a Cable... 6-9 6.5.2 Chage at a Junction... 6-10 6.5.3 Dift Velocity... 6-11 6.5.4 Resistance of a Tuncated Cone... 6-12 6.5.5 Resistance of a Hollow Cylinde... 6-13 6.6 Conceptual Questions... 6-14 6.7 Additional Poblems... 6-14 6.7.1 Cuent and Cuent Density... 6-14 6.7.2 Powe Loss and Ohm s Law... 6-14 6.7.3 Resistance of a Cone... 6-15 6.7.4 Cuent Density and Dift Speed... 6-15 6.7.5 Cuent Sheet... 6-16 6.7.6 Resistance and Resistivity... 6-16 6.7.7 Powe, Cuent, and Voltage... 6-17 6.7.8 Chage Accumulation at the Inteface... 6-17 1 These notes ae excepted Intoduction to Electicity and Magnetism by Sen-Ben Liao, Pete Doumashkin, and John Belche, Copyight 2004, ISBN 0-536-81207-1. 6-1
Cuent and Resistance 6.1 Electic Cuent Electic cuents ae flows of electic chage. Suppose a collection of chages is moving pependicula to a suface of aea A, as shown in Figue 6.1.1. Figue 6.1.1 Chages moving though a coss section. The electic cuent is defined to be the ate at which chages flow acoss any cosssectional aea. If an amount of chage ΔQ passes though a suface in a time inteval Δt, then the aveage cuent I avg is given by I avg = ΔQ (6.1.1) Δt The SI unit of cuent is the ampee (A), with 1 A = 1 coulomb/sec. Common cuents ange fom mega-ampees in lightning to nano-ampees in you neves. In the limit Δt 0, the instantaneous cuent I may be defined as I = dq (6.1.2) dt Since flow has a diection, we have implicitly intoduced a convention that the diection of cuent coesponds to the diection in which positive chages ae flowing. The flowing chages inside wies ae negatively chaged electons that move in the opposite diection of the cuent. Electic cuents flow in conductos: solids (metals, semiconductos), liquids (electolytes, ionized) and gases (ionized), but the flow is impeded in nonconductos o insulatos. 6.1.1 Cuent Density To elate cuent, a macoscopic quantity, to the micoscopic motion of the chages, let s examine a conducto of coss-sectional aea A, as shown in Figue 6.1.2. 6-2
Figue 6.1.2 A micoscopic pictue of cuent flowing in a conducto. Let the total cuent though a suface be witten as I = J da (6.1.3) whee J is the cuent density (the SI unit of cuent density ae A/m 2 ). If q is the chage of each caie, and n is the numbe of chage caies pe unit volume, the total amount of chage in this section is then Δ Q= q( na Δx). Suppose that the chage caies move with a speed v d ; then the displacement in a time inteval Δt will be Δ x = vd Δ t, which implies I = ΔQ avg = nqv d A (6.1.4) Δt The speed v d at which the chage caies ae moving is known as the dift speed. Physically, v d is the aveage speed of the chage caies inside a conducto when an extenal electic field is applied. Actually an electon inside the conducto does not tavel in a staight line; instead, its path is athe eatic, as shown in Figue 6.1.3. Figue 6.1.3 Motion of an electon in a conducto. Fom the above equations, the cuent density J can be witten as J =nqv d (6.1.5) Thus, we see that J and v d point in the same diection fo positive chage caies, in opposite diections fo negative chage caies. 6-3
To find the dift velocity of the electons, we fist note that an electon in the conducto expeiences an electic foce F e = ee which gives an acceleation F e ee a = = (6.1.6) m m e Let the velocity of a given electon immediate afte a collision be v i. The velocity of the electon immediately befoe the next collision is then given by e ee v f = v i + at = v i t (6.1.7) m e whee t is the time taveled. The aveage of v f ove all time intevals is ee v f = v i t (6.1.8) m e which is equal to the dift velocity v d. Since in the absence of electic field, the velocity of the electon is completely andom, it follows that v i = 0. If τ = t is the aveage chaacteistic time between successive collisions (the mean fee time), we have The cuent density in Eq. (6.1.5) becomes ee v d = = τ (6.1.9) v f m e ee ne 2 τ J = nev d = ne τ = E (6.1.10) m e m e Note that J and E will be in the same diection fo eithe negative o positive chage caies. 6.2 Ohm s Law In many mateials, the cuent density is linealy dependent on the extenal electic field E. Thei elation is usually expessed as J = σ E (6.2.1) 6-4
whee σ is called the conductivity of the mateial. The above equation is known as the (micoscopic) Ohm s law. A mateial that obeys this elation is said to be ohmic; othewise, the mateial is non-ohmic. Compaing Eq. (6.2.1) with Eq. (6.1.10), we see that the conductivity can be expessed as ne 2 τ σ = (6.2.2) m e To obtain a moe useful fom of Ohm s law fo pactical applications, conside a segment of staight wie of length l and coss-sectional aea A, as shown in Figue 6.2.1. Figue 6.2.1 A unifom conducto of length l and potential diffeence Δ V = V b V a. Suppose a potential diffeence Δ V = V b V a is applied between the ends of the wie, ceating an electic field E and a cuent I. Assuming E to be unifom, we then have The cuent density can then be witten as V V b V a = b Δ = E d s = El (6.2.3) a With J whee = I / A, the potential diffeence becomes is the esistance of the conducto. The equation J = σ E = σ ΔV (6.2.4) l Δ l V = l J = σ σ A I = RI (6.2.5) ΔV l R = = (6.2.6) I σ A 6-5
Δ V = IR (6.2.7) is the macoscopic vesion of the Ohm s law. The SI unit of R is the ohm (Ω, Geek lette Omega), whee 1V 1 Ω (6.2.8) 1A Once again, a mateial that obeys the above elation is ohmic, and non-ohmic if the elation is not obeyed. Most metals, with good conductivity and low esistivity, ae ohmic. We shall focus mainly on ohmic mateials. Figue 6.2.2 Ohmic vs. Non-ohmic behavio. The esistivity ρ of a mateial is defined as the ecipocal of conductivity, 1 m ρ = = e (6.2.9) σ ne 2 τ Fom the above equations, we see that ρ can be elated to the esistance R of an object by o E ΔV / l RA ρ = = = J I / A l ρl R = (6.2.10) A The esistivity of a mateial actually vaies with tempeatue T. Fo metals, the vaiation is linea ove a lage ange of T: ρ = ρ [1+α( T T ) 0 0 ] (6.2.11) whee α is the tempeatue coefficient of esistivity. Typical values of ρ, σ and α (at 20 C) fo diffeent types of mateials ae given in the Table below. 6-6
Mateial Elements Silve Resistivity ρ ( Ω m ) 1.59 10 8 Conductivity σ ( Ω m) 1 Tempeatue Coefficient α (C) 1 7 6.29 10 0.0038 8 Coppe 1.72 10 5.81 10 7 0.0039 8 Aluminum 2.82 10 8 Tungsten 5.6 10 7 3.55 10 0.0039 7 1.8 10 0.0045 Ion 8 10.0 10 7 1.0 10 0.0050 Platinum 8 10.6 10 7 1.0 10 0.0039 Alloys Bass 7 10 8 7 1.4 10 0.002 Manganin 8 44 10 7 0.23 10 1.0 10 5 Nichome 8 100 10 0.1 10 7 0.0004 Semiconductos Cabon (gaphite) 3.5 10 5 4 2.9 10 0.0005 Gemanium (pue) 0.46 2.2 0.048 Silicon (pue) 640 3 1.6 10 0.075 Insulatos Glass 10 14 10 10 14 10 10 10 Sulfu 10 15 10 15 16 Quatz (fused) 75 10 1.33 10 18 6.3 Electical Enegy and Powe Conside a cicuit consisting of a battey and a esisto with esistance R (Figue 6.3.1). Let the potential diffeence between two points a and b be Δ V = V b Va > 0. If a chage Δq is moved fom a though the battey, its electic potential enegy is inceased by Δ U =ΔqΔ V. On the othe hand, as the chage moves acoss the esisto, the potential enegy is deceased due to collisions with atoms in the esisto. If we neglect the intenal esistance of the battey and the connecting wies, upon etuning to a the potential enegy of Δq emains unchanged. Figue 6.3.1 A cicuit consisting of a battey and a esisto of esistance R. 6-7
Thus, the ate of enegy loss though the esisto is given by P = Δ U Δ t = Δq V I V (6.3.1) Δ t Δ = Δ This is pecisely the powe supplied by the battey. Using Δ V = IR, one may ewite the above equation as (ΔV ) 2 2 P= I R = (6.3.2) R 6.4 Summay The electic cuent is defined as: dq I = dt The aveage cuent in a conducto is I avg = nqv A d whee n is the numbe density of the chage caies, q is the chage each caie has, v d is the dift speed, and A is the coss-sectional aea. The cuent density J though the coss sectional aea of the wie is J = nqv Micoscopic Ohm s law: the cuent density is popotional to the electic field, and the constant of popotionality is called conductivity σ : d J = σ E The ecipocal of conductivity σ is called esistivity ρ : 1 ρ = σ Macoscopic Ohm s law: The esistance R of a conducto is the atio of the potential diffeence ΔV between the two ends of the conducto and the cuent I: 6-8
Resistance is elated to esistivity by ΔV R = I ρl R = A whee l is the length and A is the coss-sectional aea of the conducto. The dift velocity of an electon in the conducto is ee v d = τ m e whee m e is the mass of an electon, and τ is the aveage time between successive collisions. The esistivity of a metal is elated to τ by e ρ = 1 = m 2 σ ne τ The tempeatue vaiation of esistivity of a conducto is ρ = ρ0 1+α (T T 0 ) whee α is the tempeatue coefficient of esistivity. Powe, o ate at which enegy is deliveed to the esisto is 6.5 Solved Poblems P= IΔ V = I R = 2 ( ΔV ) 2 R 6.5.1 Resistivity of a Cable A 3000-km long cable consists of seven coppe wies, each of diamete 0.73 mm, bundled togethe and suounded by an insulating sheath. Calculate the esistance of the cable. Use 3 10 6 Ω cm fo the esistivity of the coppe. 6-9
Solution: The esistance R of a conducto is elated to the esistivity ρ by R = ρl/ A, whee l and A ae the length of the conducto and the coss-sectional aea, espectively. Since the cable consists of N = 7 coppe wies, the total coss sectional aea is 2 A N = N π d 2 = 7 π (0.073cm) 2 = π 4 4 The esistance then becomes 6 8 )( ) ρl (3 10 Ω cm 3 10 cm 4 R = = = 3.1 10 Ω A 7π (0.073cm) 2 / 4 6.5.2 Chage at a Junction Show that the total amount of chage at the junction of the two mateials in Figue 6.5.1 1 1 is ε 0 I(σ 2 σ 1 ), whee I is the cuent flowing though the junction, andσ 1 and σ 2 ae the conductivities fo the two mateials. Solution: Figue 6.5.1 Chage at a junction. In a steady state of cuent flow, the nomal component of the cuent density J must be the same on both sides of the junction. Since J = σ E, we have σ E = σ E 1 1 2 2 o E 2 = σ 1 E 1 σ 2 Let the chage on the inteface be q in, we have, fom the Gauss s law: o d = (E E ) A = q in E A 2 1 S ε 0 6-10
q in E 2 E 1 = A ε Substituting the expession fo E 2 fom above then yields Since the cuent is I = JA = (σ E σ 1 1 1 q in = ε 0 AE 1 1 = ε 0 Aσ 1 E 1 σ 2 σ 2 σ 1 1 1 ) 0 A, the amount of chage on the inteface becomes 1 1 q in = ε 0 I σ 2 σ 1 6.5.3 Dift Velocity The esistivity of seawate is about 25 Ω cm. The chage caies ae chiefly Na + and Cl ions, and of each thee ae about 3 10 20 3 / cm. If we fill a plastic tube 2 metes long with seawate and connect a 12-volt battey to the electodes at each end, what is the esulting aveage dift velocity of the ions, in cm/s? Solution: The cuent in a conducto of coss sectional aea A is elated to the dift speed v d of the chage caies by I = enav d whee n is the numbe of chages pe unit volume. We can then ewite the Ohm s law as which yields V = IR = (neav d ) ρl = nev d ρl A V v d = neρl Substituting the values, we have 12V 5 V cm 5 cm v = d 20 3 19 (6 10 /cm )( 1.6 10 C)(25Ω cm )( 200cm ) = 2.5 10 C Ω = 2.5 10 s 6-11
In conveting the units we have used V V 1 A 1 = = = s Ω C Ω C C 6.5.4 Resistance of a Tuncated Cone Conside a mateial of esistivity ρ in a shape of a tuncated cone of altitude h, and adii a and b, fo the ight and the left ends, espectively, as shown in the Figue 6.5.2. Figue 6.5.2 A tuncated Cone. Assuming that the cuent is distibuted unifomly thoughout the coss-section of the cone, what is the esistance between the two ends? Solution: Conside a thin disk of adius at a distance x fom the left end. Fom the figue shown on the ight, we have b b a = x h o = (a b) x + b h Since esistance R is elated to esistivity ρ by R = ρl/ A, whee l is the length of the conducto and A is the coss section, the contibution to the esistance fom the disk having a thickness dy is ρ dx ρ dx dr = = π 2 π[b + (a b) x / h ] 2 6-12
Staightfowad integation then yields h ρ dx ρh R = = 0 2 π[b + (a b) x / h ] π ab whee we have used du 1 = (αu + β ) 2 α( α u + β ) Note that if b= a, Eq. (6.2.9) is epoduced. 6.5.5 Resistance of a Hollow Cylinde Conside a hollow cylinde of length L and inne adius a and oute adius b, as shown in Figue 6.5.3. The mateial has esistivity ρ. Figue 6.5.3 A hollow cylinde. (a) Suppose a potential diffeence is applied between the ends of the cylinde and poduces a cuent flowing paallel to the axis. What is the esistance measued? (b) If instead the potential diffeence is applied between the inne and oute sufaces so that cuent flows adially outwad, what is the esistance measued? Solution: (a) When a potential diffeence is applied between the ends of the cylinde, cuent flows paallel to the axis. In this case, the coss-sectional aea is A = π (b 2 a 2 ), and the esistance is given by ρl ρl R = = 2 2 A π (b a ) 6-13
(b) Conside a diffeential element which is made up of a thin cylinde of inne adius and oute adius + d and length L. Its contibution to the esistance of the system is given by ρ dl ρ d dr = = A 2π L whee A = 2π L is the aea nomal to the diection of cuent flow. The total esistance of the system becomes b ρ d ρ b R = = ln a 2π L 2π L a 6.6 Conceptual Questions 1. Two wies A and B of cicula coss-section ae made of the same metal and have equal lengths, but the esistance of wie A is fou times geate than that of wie B. Find the atio of thei coss-sectional aeas. 2. Fom the point of view of atomic theoy, explain why the esistance of a mateial inceases as its tempeatue inceases. 3. Two conductos A and B of the same length and adius ae connected acoss the same potential diffeence. The esistance of conducto A is twice that of B. To which conducto is moe powe deliveed? 6.7 Additional Poblems 6.7.1 Cuent and Cuent Density A sphee of adius 10 mm that caies a chage of 8 nc = 8 10 9 C is whiled in a cicle at the end of an insulated sting. The otation fequency is 100π ad/s. (a) What is the basic definition of cuent in tems of chage? (b) What aveage cuent does this otating chage epesent? (c) What is the aveage cuent density ove the aea tavesed by the sphee? 6.7.2 Powe Loss and Ohm s Law A 1500 W adiant heate is constucted to opeate at 115 V. 6-14
(a) What will be the cuent in the heate? [Ans. ~10 A] (b) What is the esistance of the heating coil? [Ans. ~10 Ω] (c) How many kilocaloies ae geneated in one hou by the heate? (1 Caloie = 4.18 J) 6.7.3 Resistance of a Cone A coppe esisto of esistivity ρ is in the shape of a cylinde of adius b and length L 1 appended to a tuncated ight cicula cone of length L 2 and end adii b and a as shown in Figue 6.7.1. Figue 6.7.1 (a) What is the esistance of the cylindical potion of the esisto? (b) What is the esistance of the entie esisto? (Hint: Fo the tapeed potion, it is necessay to wite down the incemental esistance dr of a small slice, dx, of the esisto at an abitay position, x, and then to sum the slices by integation. If the tape is small, one may assume that the cuent density is unifom acoss any coss section.) (c) Show that you answe educes to the expected expession if a = b. (d) If L 1 = 100 mm, L 2 = 50 mm, a = 0.5 mm, b = 1.0 mm, what is the esistance? 6.7.4 Cuent Density and Dift Speed (a) A goup of chages, each with chage q, moves with velocity v. The numbe of paticles pe unit volume is n. What is the cuent density J of these chages, in magnitude and diection? Make sue that you answe has units of A/m 2. (b) We want to calculate how long it takes an electon to get fom a ca battey to the state moto afte the ignition switch is tuned. Assume that the cuent flowing is115 A, 2 and that the electons tavel though coppe wie with coss-sectional aea 31.2 mm and length 85.5 cm. What is the cuent density in the wie? The numbe density of the conduction electons in coppe is 8.49 10 28 /m 3. Given this numbe density and the cuent density, what is the dift speed of the electons? How long does it take fo an 6-15
electon stating at the battey to each the state moto? [Ans: 3.69 10 6 A/m 2, 2.71 10 4 m/s,52.5 min.] 6.7.5 Cuent Sheet A cuent sheet, as the name implies, is a plane containing cuents flowing in one diection in that plane. One way to constuct a sheet of cuent is by unning many paallel wies in a plane, say the yz -plane, as shown in Figue 6.7.2(a). Each of these wies caies cuent I out of the page, in the ĵ diection, with n wies pe unit length in the z-diection, as shown in Figue 6.7.2(b). Then the cuent pe unit length in the z diection is ni. We will use the symbol K to signify cuent pe unit length, so that K = nl hee. Figue 6.7.2 A cuent sheet. Anothe way to constuct a cuent sheet is to take a non-conducting sheet of chage with fixed chage pe unit aea σ and move it with some speed in the diection you want cuent to flow. Fo example, in the sketch to the left, we have a sheet of chage moving out of the page with speed v. The diection of cuent flow is out of the page. (a) Show that the magnitude of the cuent pe unit length in the z diection, K, is given by σ v. Check that this quantity has the pope dimensions of cuent pe length. This is in fact a vecto elation, K (t) = σ v ( t ), since the sense of the cuent flow is in the same diection as the velocity of the positive chages. (b) A belt tansfeing chage to the high-potential inne shell of a Van de Gaaff acceleato at the ate of 2.83 mc/s. If the width of the belt caying the chage is 50 cm and the belt tavels at a speed of 30 m/s, what is the suface chage density on the belt? [Ans: 189 μc/m 2 ] 6.7.6 Resistance and Resistivity A wie with a esistance of 6.0 Ω is dawn out though a die so that its new length is thee times its oiginal length. Find the esistance of the longe wie, assuming that the 6-16
esistivity and density of the mateial ae not changed duing the dawing pocess. [Ans: 54 Ω]. 6.7.7 Powe, Cuent, and Voltage A 100-W light bulb is plugged into a standad 120-V outlet. (a) How much does it cost pe month (31 days) to leave the light tuned on? Assume electicity costs 6 cents pe kw h. (b) What is the esistance of the bulb? (c) What is the cuent in the bulb? [Ans: (a) $4.46; (b) 144 Ω; (c) 0.833 A]. 6.7.8 Chage Accumulation at the Inteface Figue 6.7.3 shows a thee-laye sandwich made of two esistive mateials with esistivities ρ 1 and ρ 2. Fom left to ight, we have a laye of mateial with esistivity ρ 1 of width d /3, followed by a laye of mateial with esistivity ρ 2, also of width d /3, followed by anothe laye of the fist mateial with esistivity ρ 1, again of width d /3. Figue 6.7.3 Chage accumulation at inteface. The coss-sectional aea of all of these mateials is A. The esistive sandwich is bounded on eithe side by metallic conductos (black egions). Using a battey (not shown), we maintain a potential diffeence V acoss the entie sandwich, between the metallic conductos. The left side of the sandwich is at the highe potential (i.e., the electic fields point fom left to ight). Thee ae fou intefaces between the vaious mateials and the conductos, which we label a though d, as indicated on the sketch. A steady cuent I flows though this sandwich fom left to ight, coesponding to a cuent density J = I / A. (a) What ae the electic fields E 1 and E 2 in the two diffeent dielectic mateials? To obtain these fields, assume that the cuent density is the same in evey laye. Why must this be tue? [Ans: All fields point to the ight, E 1 = ρ 1 I / A, E 2 = ρ 2 I / A ; the cuent densities must be the same in a steady state, othewise thee would be a continuous buildup of chage at the intefaces to unlimited values.] 6-17
(b) What is the total esistance R of this sandwich? Show that you expession educes to the expected esult if ρ 1 = ρ 2 = ρ. [Ans: R = d (2 ρ 1 + ρ 2 )/3 A ; if ρ 1 = ρ 2 = ρ, then R = d ρ / A, as expected.] (c) As we move fom ight to left, what ae the changes in potential acoss the thee layes, in tems of V and the esistivities? [Ans: V ρ 1 / 2 V ρ 1 /( 2ρ 1 + ρ 2 ), summing to a total potential dop of V, as equied]. ( ρ 1 + ρ 2 ),V ρ 2 /( 2ρ 1 + ρ 2 ), (d) What ae the chages pe unit aea, σ a though σ d, at the intefaces? Use Gauss's Law and assume that the electic field in the conducting caps is zeo. [Ans: σ a = σ d = 3ε 0 V ρ 1 / d (2 ρ 1 + ρ 2 ), σ b = σ c = 3ε 0 V (ρ 2 ρ 1 )/ d (2 ρ 1 + ρ 2 ).] (e) Conside the limit ρ 2 ρ 1. What do you answes above educe to in this limit? 6-18
MIT OpenCouseWae http://ocw.mit.edu 8.02SC Physics II: Electicity and Magnetism Fall 2010 Fo infomation about citing these mateials o ou Tems of Use, visit: http://ocw.mit.edu/tems.