Physics 2B Chapter 23 Notes  Faraday s Law & Inductors Spring 2018


 Domenic Fox
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1 Michael Faraday lived in the Lndn area frm 1791 t He was 29 years ld when Hand Oersted, in 1820, accidentally discvered that electric current creates magnetic field. Thrugh empirical bservatin and a lt f trial and errr, arund 1830 Faraday wuld make ne f the mst imprtant discveries f the century: that, under certain circumstances, a magnetic field can create current. What are these certain circumstances? Faraday fund that a static magnetic field prduces n induced current, but changes in a magnetic field passing thrugh a cil f wire wuld induce a current in the cil. The key was that the magnetic field thrugh the cil had t change smehw. Faraday s Law f Inductin summarized the effect by first defining magnetic flux: the prduct f an area and the magnitude f the magnetic field that passes thrugh the area. We can write the magnetic flux as: φ B Bcsθ B where B and are the magnitudes f the magnetic field and area, respectively, and θ is the angle between the directins f B and. Faraday discvered that any change in the magnetic flux will induce a ptential arund the lp; if the lp is a cnductr, a crrespnding current will als be induced. Faraday s Law prvides an expressin fr the induced ptential; if a current is t be induced, the cnductr at issue must have sme resistance and we simply cmbine Faraday s Law with Ohm s Law (frm Chapter 21) t determine the induced current (caused by the induced ptential.) ind dφb N d( Bcsθ ) N Faraday s Law f Inductin few imprtant ntes: The N in frnt takes int cnsideratin the pssibility that the area might be that f a cil f N turns. In such a case, the ttal induced ptential f all the turns cmbined (as they are in series) is simply N times the induced ptential f ne turn. Fr mst prblems, N will be ne. The minus sign in frnt indicates the directin f the induced ptential (and crrespnding induced current, where applicable.) Essentially the minus sign tells us that the induced ptential will be in the directin ppsite the change (either increase r decrease) f the magnetic flux. See belw fr an explanatin, via Lenz Law, fr hw t determine the directin f the induced ptential. T induce a ptential, any r all f B r csθ can change, althugh mst prblems will keep tw f these cnstant and change nly f the three. See belw fr mre details. Page 1 f 6
2 Three ersins f Faraday s Law Faraday s Law tells us that if the magnetic flux thrugh a lp changes, a ptential will be induced arund the lp. There are three ways that the magnetic flux can change: B can change, can change, r the angle between B and can change. If B, the magnetic field, changes: If a prblem presents yu with a changing magnetic field, then the area thrugh which it passes and the angle between the area and the magnetic field will bth be cnstant. The prblem will either suggest that the magnetic field varies cntinuusly, and will give yu a way t determine the rate, i.e. db/, either by prviding it directly r prviding a functin f which yu can take the derivative with respect t time. Or yu will be asked t find the average induced ptential by replacing the rate f change with the change in B (i.e. final minus initial) divided by the time interval. We can simplify Faraday s Law t then read: ind avg db Ncsθ B changes cntinuusly B N θ t cs B changes frm initial t final ver a specified time interval If, the area, changes: If a prblem presents yu with a changing area, then the magnetic field and the angle between the area and the magnetic field will bth be cnstant. Just as with the changing magnetic field scenari abve, yu will either be prvided a rate f change f the area, r yu will be asked t find the average induced ptential by cnsidering the change in the area ver a given time interval. We can simplify Faraday s Law t then read: ind avg d NB csθ changes cntinuusly NB θ t cs changes frm initial t final ver a specified time interval If θ, the angle, changes: If a prblem presents yu with a changing angle, then the magnetic field and the angle between the area and the magnetic field will bth be cnstant. If the angle changes cntinuusly, then we can assume the Page 2 f 6
3 cil (r lp) is rtating at a cnstant rate. We can define this rate as an angular velcity (Chapter 10) given by the symbl ω, and the angle at any given time is θ ωt. (Nte that we simply define angle zer t be when time is zer.) With this definitin, Faraday s Law becmes: ind d(csωt) NB NBω( sinωt) NBω sinθ nd s ur tw pssible expressins fr a changing angle are: NBω sinθ ind θ changes cntinuusly, a cil rtating with ang velcity ω (csθ ) avg NB t θ changes frm initial t final ver a specified time interval The six versins f Faraday s Law, in the three bxes abve, are what yu will use t tackle the prblems in Chapter 23. Each prblem will simply require that yu evaluate the circumstances presented in each prblem and chse the apprpriate expressin fr thse circumstances. Directin f the Induced Ptential r Current: Lenz Law Lenz Law was created by Heinrich Lenz in 1834 as a cmplement t Faraday s Law f Inductin. Lenz Law is a statement f the directin f the ptential, and current (if applicable), induced in a lp r cil. It states simply that the directin f the induced current will be such that it will ppse the change in the magnetic flux (and thus the minus sign in Faraday s Law.) But what des this mean, t ppse? T make sense f this, it is imprtant t realize that the induced current is an actual current that has all the prperties f a current that we explred in Chapter 22. That is, the induced current creates its wn magnetic field. The key t understanding Lenz Law is t realize that the magnetic field created by the induced current will be in such a directin as t ppse the increase f decrease f the magnetic flux that was respnsible fr inducing the current in the first place. nd if yu knw the directin f the magnetic field created by the induced current, yu shuld be able t determine the directin f the induced current. The prcess fr using Lenz Law is simply three steps: 1. Did the magnetic flux increase r decrease, and which directin? 2. What directin wuld a magnetic field need t be created t ppse the change identified in Step 1? 3. Determine the directin f the induced current that created such a magnetic field. Page 3 f 6
4 Inductrs n inductr is simply a lng, thin cil f wire... that is, a slenid! We can repurpse the picture frm the Chapter 22 ntes: Slenid Inductr with N turns, length L and crsssectin area n inductr and a slenid are the same thing. Why the tw different names? By cnventin we call the cil a slenid if we are using it t create a magnetic field inside (fr whatever reasn... slenids are ften used in mechanical devices, like dr lcks, by which the magnetic field in the slenid alternatively attracts r repels a magnetized rd.) s a slenid, the current is typically turned n r ff t create a magnetic field that can be turned n r ff... like an electrmagnet. If we use the cil as an inductr, it has a different purpse. In this case, we expect the current in the inductr t change and the change f the current will cause a changing magnetic field in the middle f the inductr. This will create an induced ptential in each lp f the inductr; the ttal induced ptential then will be N times the induced ptential f ne lp. Using Faraday s Law, we can write: µ B NI L Magnetic field in a slenid inductr ind db csθ Ptential induced in ne lp f the inductr cil We multiply the secnd expressin by N, t accunt fr all the lps f the inductr. nd we acknwledge that the magnetic field passes directly thrugh the area f the lps, s the angle θ is zer. Finally, we substitute the expressin fr B and the result is an expressin fr the ttal ptential induced in the lps f ur inductr: Page 4 f 6
5 ind d(µ NI N / L) Or: µ N ind L 2 di t this pint we make a definitin: the inductance f the cil. Unfrtunately the symbl we use fr inductance is L, which I have already used as the length f the inductr cil. I will switch nw t use l (a script L ) fr the length f the inductr cil. Then the inductance f the cil is defined by: µ L N l 2 Inductance f an inductr cil We have a new cncept: inductance. We have t define the SI unit f inductance. If we cnsider the units f µ, the unit f and the unit f l (nte that N has n units) we can see that the units f L are: (Tm/) m 2 / m r Tm 2 / We define this as a Henry, after the merican physicist Jseph Henry (wh made imprtant cntributins t the study f inductance during the 19 th century.) Henry is abbreviated with a capital H, s: SI unit f inductance: H Tm 2 / Cnsidering that the µ in the expressin fr inductance includes a factr f 107, it shuld be fairly easy t see that creating a inductr with an inductance f ne Henry is quite a feat! The ther three factrs: the number f lps, the area f the lps r the length f the cil, wuld have t be planned s as t vercme that factr f Fr this reasn, cmmn inductrs are frequently measured in millihenries (i.e. mh), r micrhenries (i.e. µh.) We can nw rewrite the expressin fr the ptential induced acrss an inductr cil (i.e. frm ne side t the ther) as: di ind L where L is the inductance f the cil. Page 5 f 6
6 We can nw see hw inductrs cmplete ur set f elements we use in electric circuits. In Chapter 20 we fund capacitrs, in Chapter 21 we fund resistrs, and Chapter 23 we nw have fund inductrs. Capacitr: 1 q Special case? Parallel plate capacitr: C ε C d Resistr: R I Special case? Resistance f a wire: R w k Inductr: di L Special case? Inductr cil: µ L N l 2 Nte the prgressin in the expressins fr the ptential acrss a capacitr, resistr and inductr: The ptential acrss the capacitr is prprtinal t the charge, q, n the capacitr. The ptential acrss the resistr is prprtinal t the current, dq/, thrugh the resistr. The ptential acrss the inductr is prprtinal t the rate f change f the current, di/. Just as with capacitrs and resistrs, we have a symbl fr an inductr that we use in circuit diagrams: Capacitr Resistr Inductr What are inductrs used fr? If the current in a circuit tries t increase r decrease (i.e. there is a di/) then an inductr in the circuit with inhibit the change in the current in much the same way that the resistr in the RC circuit slwed dwn the rate at which charge accumulated n the capacitr. Inductrs can be useful, therefre, in simple surge cntrl circuits: if smething unexpected tries t increase the current in a circuit (a pwer surge?) then an inductr with resist the increase in current (and effectively blck the damaging effects f the surge.) Simple dimmer switches take advantage f the fact that the current in ur hmes is alternating... and therefre an inductr presents a type f resistance in such a circuit. variable inductr (yu vary it by turning a knb) can therefre act as a dimmer switch, varying the amunt f current that is allwed thrugh the switch t pwer the light attached t the switch. Page 6 f 6
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