SELF-INDUCTANCE AND INDUCTORS

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1 MISN SELF-INDUCTANCE AND INDUCTORS SELF-INDUCTANCE AND INDUCTORS by Pete Signell Michigan State Univesity 1. Intoduction A 2. Self-Inductance L a. The Definition of L b. L When the Flux is Known c. L fo a Tooidal Solenoid L/l fo a Coaxial Cable a. Physical Desciption of a Coaxial Cable b. Getting B c. Getting L/l Inductive Enegy In a Cicuit a. Enegy Needed to Set up a Cuent b. The Enegy is Recoveable c. Location of the Enegy d. Enegy Flow When a Cuent is Stopped Acknowledgments Glossay Poject PHYSNET Physics Bldg. Michigan State Univesity East Lansing, MI 1

2 ID Sheet: MISN Title: Self-Inductance and Inductos Autho: Pete Signell, Michigan State Univesity Vesion: 1/25/2001 Evaluation: Stage 0 Length: 1 h; 12 pages Input Skills: 1. Vocabulay: solenoid, tooid, inductance, induced voltage, induced cuent, induced magnetic field, Faaday-Heny law, Lenz s law (MISN-0-142). 2. Use Ampee s law to detemine the magnetic field due to a long staight cuent (MISN-0-138). Output Skills (Knowledge): K1. Vocabulay: coaxial cable, heny, inducto, self-inductance. K2. Wite down Ampee s Law and fom it deive the self-inductance of a tooidal solenoid, explicitly justifying each step. K3. Wite down Ampee s Law and fom it deive the self-inductance pe unit length of a coaxial cable, explicitly justifying each step. K4. Stating fom the elation between powe, voltage and cuent in a steady state cicuit, deive the enegy stoed in the electic field of an inducto. K5. Descibe the flow of enegy: (a) when the cuent though an inducto is inceased; (b) when the cuent though an inducto is vey gadually deceased; and (c) when the cuent though an inducto is quickly deceased. Post-Options: 1. Two Element DC-Diven LRC Cicuits (MISN-0-151). 2. Velocity of a Signal in a Coaxial Cable (MISN-0-150). 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) Fo ou libeal use policies see: 3 4

3 MISN SELF-INDUCTANCE AND INDUCTORS by Pete Signell Michigan State Univesity 1. Intoduction A solenoid o a tooid, sometimes of miniatue size, is used in electonic cicuits to: (1) slow the ate of change of electic cuent; (2) tune a cicuit to a paticula oscillational fequency; o (3) contol the speed of tansmission of signals. In these applications one is making use of the fact that a change in the cuent going though the device poduces a change in the associated magnetic field and that in tun induces a cuent that opposes the change in the oiginal cuent. A device used in that manne is called an inducto and the stength of its change-opposing chaacte is called its self-inductance o simply its inductance. This (self) inductance is measued in the S.I. unit called the heny, so a cicuit designe may specify an inducto of, say, 35 milli-henies to achieve one of the thee above-mentioned aims. 2. Self-Inductance L 2a. The Definition of L. Inductance in geneal includes potentials and cuents induced in one conducto by a time-changing cuent in anothe conducto, but self-inductance efes to potentials and cuents that ae induced in a single conducto by its own time-changing cuent. A device used fo this pupose usually has the shape of a solenoid o a tooid. A cuent flowing though an inducto of couse sets up a magnetic field so changes in the cuent poduce changes in the magnetic field. Such changes poduce an induced voltage dop in the inducto, a voltage dop that opposes the change in the cuent. The magnitude of the induced voltage is popotional to the time-ate-of-change of the cuent, as we have seen, 1 so we wite: V ind = L di dt. (1) 1 See Magnetic Inductance, MISN MISN A Figue 1. A tooidal solenoid. Hee L is a popotionality constant that depends on the geomety of the inducto and the inducto s mateial: it is called the inducto s inductance. An inducto s inductance can be much enhanced by placing its loops of wie aound a magnetic mateial such as ion. 2b. L When the Flux is Known. If one knows the flux though the suface bounded by a cicuit, then the self-inductance can be detemined by integating Eq. (1) to get: Φ = L I. (2) Fo example, if the flux enclosed by a loop is T m 2 when the cuent in the loop is 1.3 A, then the inductance of the loop is L = 20 mh. If the inducto is alteed to contain 15 successive loops glued togethe, then the flux met by the cuent will be inceased by a facto of 15 and so will L. 2c. L fo a Tooidal Solenoid. To find the self-inductance of a tooidal solenoid, we use Ampee s law and we daw the integation loop as a cicle of adius fom the cente of the tooid (see Fig. 1). If is less than the inne adius of the tooid, thee is zeo cuent going though any suface bounded by the integation loop. If is between the inne and oute adii of the tooid the the cuent going though the enclosed suface is NI whee N is the numbe of tuns of wie caying the cuent I. If is geate than the oute adius of the tooid, thee is again zeo net cuent going though any suface bounded by the integation loop (the cuent going one diection though the suface is exactly canceled by the cuent going the othe way though the suface). Then by Ampee s law, the magnetic field inside the solenoidal loops is: B() = 2 k m N I. (3) 5 6

4 MISN MISN dielectic (usually white, flexible) baided wie sheath V 0 cable R Figue 3. Longitudinal view of a coaxial cable in a hypothetical cicuit. cente wie (solid) plastic skin Figue 2. Coss-section of a coaxial cable. If the adius of each loop is much smalle than the tooidal adius, then will vay little ove the coss-sectional aea A of each loop and we can take as a constant. To get the flux in a single loop of the winding, we need meely multiply the B by the loop aea A. Then fo the N windings in the inducto: Φ = N A B = 2 k m N 2 I A. (4) Then fom Eq. (2): L = 2 k m N 2 A. (5) Typical numbes fo a cicuit tooid will give an extemely small inductance unless the flux is enhanced though the use of an ion coe inside the solenoidal loops. This enhancement may be by a facto of five thousand o moe. 3. L/l fo a Coaxial Cable 3a. Physical Desciption of a Coaxial Cable. Physically, a coaxial cable looks like a fat ound wie: a good example is the cable that feeds television pogams to TV sets fom a cable company. If you wee to cut though such a cable you would see a cental conducting wie suounded by a dielectic mateial (usually flexible white plastic). This insulating mateial is suounded by a cylindical sheath woven fom conducting wies (woven to make it flexible). That oute conducto is, in tun, coveed by a thin skin of flexible insulating plastic (see Fig. 2). Cuent flows down one conducto and exactly the same amount of cuent flows in the opposite diection in the othe conducto. One can imagine a voltage souce connected between the inne and oute conductos at one end of the cable and a esisto connected between them at the othe end, as in Fig. 3. We will assume that the cuent taveling the inne wie is along its suface, at a adius i (a good assumption fo high-fequency waves). We will call the adius of the oute conducto o. 3b. Getting B. Fo self-inductance we need flux pe unit cuent, and fo flux we need the magnetic field. Fo a coaxial cable the entie magnetic field is between the two conductos. That is easily seen because by Ampee s law thee is zeo net cuent cossing a coss-sectional aea lage than the outside conducto (emembe that the cuents in the two conductos ae equal but opposite in diection). Fo the egion between the two conductos, Ampee s law immediately shows that the magnetic field is simple that of the inne conducto. Fo a long staight wie it is: 2 B = 2 k m I and of couse the diection of the field is eveywhee pependicula to the outwad cylindical adius fom the inne wie. 3c. Getting L/l. To get the flux in ou coaxial cable we must integate the component of the magnetic field nomal to a suface bounded by the loop of electical cuent. To make it easy we choose a suface that is Apply Ampee s law in you head o see Magnetic Fields fom Cuents, MISN- A B D C, Figue 4. A longitudinal view of the coaxial cable showing the integation suface ABCD fo obtaining the flux. 7 8

5 MISN adial, unning between the two conductos and along the length of the cable (see Fig. 4). Now the element of flux at a paticula adius is just the value of B thee times the element of aea at that adius: ( ) 2 km I dφ = (B) da = l d. whee l is the length of the cable. Integating, o ( ) 2 km I Φ = dφ = l d = 2 k m I l ln ( o / i ). i Finally, then, the self-inductance pe unit length of cable is: L/l = Φ/l I = 2 k m ln ( o / i ). (6) The total inductance fo any paticula piece of cable can be obtained by multiplying L/l by the piece s length. 4. Inductive Enegy In a Cicuit 4a. Enegy Needed to Set up a Cuent. When a cicuit switch is closed, stating a flow of cuent though a cicuit, an inducto in the path of the cuent esists the ise of the cuent fom zeo by developing a counte-voltage dop (a voltage ise). The souce of voltage in the cicuit must push the cuent past this voltage ise, doing wok equal to the voltage ise times the amount of chage pushed though it. The powe expended (the enegy pe unit time) is just the cuent (the chage pe unit time) times the self-induced voltage: P = I L di dt. Using P = de/dt we can easily integate both sides of: to get: de = I L di E = 1 2 L I2. (7) MISN b. The Enegy is Recoveable. The enegy expended in setting up a cuent in an inducto is ecoveable if the cicuit s voltage souce is emoved. Assuming thee is still a complete cicuit without the voltage souce, the inducto will keep the cuent flowing until the enegy of Eq. (7) has been completely dissipated in the cicuit s esistances o pehaps tansfeed to a capacitance fo stoage thee. 4c. Location of the Enegy. We descibe the enegy of Eq. (7) as being stoed in the magnetic field of the inducto. That stoed enegy was zeo in the beginning of ou example, when the cicuit cuent was zeo so the inducto s magnetic field was zeo. As the cuent, hence the inducto s magnetic field, inceased the enegy in the magnetic field inceased as the cicuit s voltage souce supplied enegy to the field. If the cuent became steady, the inducto s magnetic field became steady along with it and thee was no longe a tansfe of enegy fom the voltage souce to the inducto s field. Howeve, the enegy stoed in the inducto s magnetic field stays stoed thee. 4d. Enegy Flow When a Cuent is Stopped. Suppose a steady cuent is flowing in a cicuit containing an inducto, and then one opens a switch so cuent can pesumably no longe flow: what happens to the enegy stoed in the inducto s magnetic field? The answe is that as the switch is opened the cuent will dop quickly, ceating a lage induced voltage that ionizes the ai acoss the switch gap and thus causes an electic ac. The enegy stoed in the inducto s magnetic field is thus dissipated in chemical and heat enegy in beaking down the ai in the switch gap and in buning the contact points in the switch and in buning anything else in the vicinity. In fact, the stoed enegy in a lage inducto can be extemely dangeous to anyone attempting to stop the cuent in a huy. Acknowledgments Pepaation of this module was suppoted in pat by the National Science Foundation, Division of Science Education Development and Reseach, though Gant #SED to Michigan State Univesity. This is the enegy that must be expended by the cicuit s enegy souce in ode to aise the cicuit cuent fom zeo to the value I. 9 10

6 MISN MISN ME-1 Glossay co-axial cable: a cicuit element that looks like a long fat plasticcoveed wie, containing in successive cylindical layes: an inne solidwie conducto, a suounding flexible dielectic, and an oute baidedwie sheath. The cicuit s cuent goes down one conducto (solid o sheath) and back the othe. heny: the SI unit of self-inductance, abbeviated H and defined to be an ohm-second. Thus: H Ω s = V s A 1. inducto: a cicuit element whose pupose is to povide selfinductance, an electical cicuit analog of mechanical inetia (mass). An inducto is usually in the shape of a solenoid o a tooid. The inductance of an inducto depends on the geomety of the inducto and the magnetic susceptibility of the mateials of which the inducto is constucted. Inductos in electonic cicuits typically ae in the mh ange. MODEL EXAM 1. See Output Skills K1-K5 in this module s ID Sheet. As usual, Vocabulay means defining the wods as well as being able to use them popely. Bief Answes: 1. See this module s text. self-inductance: the negative of the induced voltage aound a loop divided by the time-ate-of-change of magnetic flux though any suface bounded by that loop: L = V induced / Φ. The minus sign shows that the induced voltage opposes the change in the flux. The SI unit of inductance is the heny (which see)

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