Inducton and Oscllatons Ch. 3: Faraday s Law Ch. 3: AC Crcuts Induced EMF: Faraday s Law Tme-dependent B creates nduced E In partcular: A changng magnetc flux creates an emf n a crcut: Ammeter or voltmeter. Electromagnetc Inducton Current n secondary crcut can be produced by a changng current n prmary crcut. Demonstratons EMF nduced n a col by movng a bar magnet Ammeter or voltmeter. Applcaton: Transformer EMF nduced n a secondary col by changng current n prmary col Sorry, we can t do t n ths packed room but here s the essence of t EMF nduced n a col by movng a bar magnet EMF nduced n a secondary col by changng current n prmary col EFM depends on how strong magnet and how fast we move n/out A
Magnetc Flux e defne magnetc flux Φ exactly as we defned the flux of the electrc feld. The dea s the number of lnes of B that pass through an area. Φ = B da Smple case #: unform B, surface: Φ = Smple case #: surface s closed: Φ = BA Faraday s Law = The emf nduced n any loop or crcut s equal to the negatve rate of change of the magnetc flux through that loop. oltmeter reang gves rate of change of the number of lnes lnkng the loop. Changng Magnetc Flux Φ = B da How can we get a tme-changng flux, so that =? Change the feld: Φ = B(t) A Change the area: Φ = B A(t) Change the angle: Φ = B A cos θ(t) Example A crcle of raus cm n the xy plane s formed by a wre and a 3-ohm resstor. A unform magnetc feld s n the z recton; ts magntude decreases stealy from.8 tesla to n a tme of 4 seconds. hat emf s generated? A = = db.8t 4 s = π r.3m = =.T / s Φ db = A = (.3)(.) =.6 d 3 Lenz s Law = The recton of the nduced emf s such as to create a current whch wll oppose the change n the flux. I push a rod along metal rals through a unform magnetc feld. Example (a) hat emf s generated? Moton as shown produces clockwse current whch makes B feld opposng the ncrease. (b) hat current wll flow? (c) hat power must I supply?
Example a L = cm = 3. m/s B =.5 T (a) hat emf s generated? da = = dx L = Lv =.6 m / s da = B =. 5.6 = 3 m Example b esstance of bar: = 5 Ω (b) hat current wll flow? 3 3 = = = ma 5Ω hch recton does current flow? Forget the mnus sgn. Use Lenz s Law! Flux s ncreasng outward. Therefore current wll resst that change by flowng clockwse. Example c Faraday s Law: General Form (c) hat power must I supply? Magnetc force: F = L B Power: P = Fv = ( F =...5 = N)(3m / s) = 6 Check Joule heatng: P = = 6 N C E ds = d S B da Φ Inductance Inductors For any col of wre, there s a flux Φ through the col, whch s proportonal to the current. If that changes, Faraday s Law requres an emf nduced n the col, proportonal to the rate of change of the flux. Clearly Φ and so = Defne the proportonalty constant to be the nductance L: = L If current s ncreasng, the nduced emf acts aganst the ncrease, gvng a voltage drop. If current s decreasng, the nduced emf acts aganst the decrease, gvng a voltage rse. SI unt of nductance s the henry (H).
Energy n an Inductor The energy stored n an nductor equals the work requred to set up the current. dq d = dq = = ( L ) = L = I d = L = LI So energy stored n an nductor s U = L Magnetc Feld Energy The energy stored n an nductor s contaned n the magnetc feld. The general formula for the energy densty n any magnetc feld s B u = µ Inductors and esstors oltage changes gong clockwse around ths loop: + L = Inductor gves voltage drop f current s ncreasng. L Crcuts + L L + = Same equaton as for chargng a capactor! t = Try same knd of soluton: = { e } Ths works, provded τ = L / L Summary Set swtch to poston a: t = { e } Set swtch to poston b: In ether case tme constant s: t = e τ = L / Example = (a) hat s the tme constant? = 3 3 = 5Ω L = 5 mh 5 6 τ = L / = = 3 = 3 µ s 3 5 (b) hat s current after second? t / τ 3 { e } = ( ) = 6 ma 5
Example : Problem 3-89 (a) hat happens mmeately after swtch s closed? L prevents sudden change so: = = = / So: = L = and = / L Example contnued (b) hat happens a long tme after swtch s closed? e have reached a steady state so: = L = and = So: = /, = / = +, Inducton and Oscllatons Ch. 3: Faraday s Law Ch. 3: AC Crcuts Chapter 3 Homework for Monday: Questons, 3, 7 Problems 3, 5, 9, 44 Chapter 3 Homework for Tuesday: Questons 3, 4, 7 Problems 5, 9, 39 leyplus chapters 3, 3 for Tuesday.