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March 16 Induction and Inductance Chapter 31 Review/Demo (Fig. 31-1) > > > A current can produce a B field Can a B field generate a current? Move a bar magnet in and out of loop of wire Moving magnet towards loop causes current in loop o Current disappears when magnet stops o Move magnet away from loop current again appears but in opposite direction o Faster motion produces a greater current o March 16, 2004 PHY 184 2 Review: Faraday’s law r r Φ B = ∫ B •dA > Magnetic flux through area A > dA is vector of magnitude dA that is ⊥ to the differential area, dA € > If B is uniform and ⊥ to A then > SI unit is the weber, Wb March 16, 2004 PHY 184 Φ B = BA Wb = T⋅m 2 3 Review: Faraday’s law • If B is constant within coil r r ΦB = ∫ B •dA = BAcosθ ε • We can change the magnetic flux through a loop (or coil) by o Changing magnitude of B field within coil o Changing area of coil, or portion of area within B field o Changing angle between B field and area of coil (e.g. rotating the coil) March 16, 2004 PHY 184 dΦ B = −N dt ε dB = − NA cosθ dt ε dA = − NB cosθ dt ε d (cosθ ) = − NBA dt 4 Checkpoint #2 > Three identical circular conductors in uniform B fields that are either increasing or decreasing in magnitude at identical rates. Rank according to magnitude of current induced in loop, greatest first. > > Use Lenz’s law to find direction of Bi Use right-hand rule to find direction of current March 16, 2004 PHY 184 5 INC=increase > Situation (a): o o B increases out of page, so Bi is into page From right-hand rule, induced current is clockwise March 16, 2004 PHY 184 6 INC=increase DEC=decrease > Situation (b) top: B increases into the page, so Bi is out of the page > Situation (b) bottom: B decreases out of the page, so Bi is out of the page > In both cases from the right-hand rule, induced current is counter-clockwise March 16, 2004 PHY 184 7 INC=increase DEC=decrease > Situation (c) top: o > Situation (c) bottom: o > > B decreases out of page, so Bi is out of the page B increases out of the page, so Bi is into the page Total Bi is zero and total current is zero Rank magnitude of current induced in loops a & b tie, then c March 16, 2004 PHY 184 8 Problem 31-3 (Fig.31-9) > > Loop has width W=3.0m and height H=2.0m Loop in non-uniform and varying B field ⊥ to loop and directed into the page 2 B = 4t x > > 2 Q : What is magnitude and direction of induced emf around loop at t=0.10s? Since magnitude B is changing in time, flux through the loop is changing so use Faraday’s law to calculate induced emf dΦ B ε March 16, 2004 =− PHY 184 dt 9 € Problem 31-3 (Fig.31-9) > 2 B = 4t x B is not uniform so need to calculate magnetic flux using 2 r r ΦB = ∫ B •dA > B ⊥ to plane of loop and only € changes r rin x direction B •dA = BdA = BHdx € > At time t: 3 x 2 ΦB = ∫ BHdx = 4t H ∫ 0 x dx = 4t H = 72t 3 0 2 March 16, 2004 3 3 2 PHY 184 2 10 Problem 31-3 (Fig.31-9) > Now use Faraday’s law to find the magnitude of the induced emf ε > > B = 4t 2 x 2 2 dΦ B d (72t ) = = = 144t dt dt At t=0.10s, emf = 14 V Find direction of emf by Lenz’s law o o B is increasing in time directed into the page, so Bi is in opposite direction - out of the page Right-hand rule – current (and emf) are counterclockwise March 16, 2004 PHY 184 11 Loop + magnet (Fig.31-10) > > > If you pull a loop at a constant velocity, v, through a B field, you must apply a constant force, F As you move loop to right, less area is in B field so magnetic flux decreases and current is induced in loop Magnetic flux when B is ⊥ and constant to area is Φ B = BA = BLx March 16, 2004 PHY 184 12 Loop + magnet (Fig.31-10) > Using Faraday’s law ε > dΦ B d dx = = BLx = BL dt dt dt Remember v = dx/dt so ε = BLv > > where L is the length of the loop and v is ⊥ to B field B is decreasing so Bi is in same direction (into page), so the current is clockwise March 16, 2004 PHY 184 13 Loop + magnet (Fig.31-10) > Since loop carries current through a B field there is a force given by r r r FB = iL × B > > > Use right-hand rule to find direction of FB on segments of loop in B field Find forces, F2 and F3 , cancel each other Force, F1 = iLB , opposes your force March 16, 2004 PHY 184 r r Fapp = − F1 14 Loop + magnet (Fig.31-10) ε = BLv > The circuit diagram is > With > Then > And March 16, 2004 ε = iR ε BLv i= = R R 2 2 B Lv F1 = iLB = R PHY 184 15 Loop + magnet (Fig.31-10) > > What happens if we push the wire in? i B is increasing so Bi is in the opposite direction (out of page), so the current is counterclockwise. v March 16, 2004 PHY 184 16 Inductance (19) > Checkpoint #3 – Four wire loops with edge lengths of either L or 2L. All loops move through uniform B field at same velocity. Rank the four loops according to maximum magnitude of induced emf, greatest first. L 2L ε = BLv March 16, 2004 c & d tie, then a & b tie PHY 184 17 Loop + magnet (Fig.31-10) > Energy is conserved - so where does the work you do moving the loop in and out go? > The current flowing through the resistance produces heat at the rate 2 2 2 B Lv P=i R= R 2 March 16, 2004 since PHY 184 BLv i= R 18 Eddy currents (Fig. 31-12) > > > Instead of a loop of wire, what happens when a bulk piece of metal moves through a B field? Free electrons in metal move in circles as if caught in a whirlpool called eddy currents A metal plate swinging through a B field will generate eddy currents March 16, 2004 PHY 184 19 Eddy currents (Fig. 31-12) > > > Eddy currents will oppose the change that caused them – Lenz’s law Induced eddy currents will always produce a retarding force when plate enters or leaves B field causing the plate to come to rest Cutting slots in metal plate will greatly reduce the eddy currents March 16, 2004 PHY 184 20 Eddy currents > > > > > > Induction and eddy currents are used for braking systems on some subways and rapid transit cars Moving vehicle has electromagnet (e.g. solenoid) which is positioned near steel rails Current in electromagnet generates B field Relative motion of B field to rails induces eddy currents in rails Eddy currents produce a drag force on the moving vehicle Eddy currents decrease steadily as car slows giving a smooth stop March 16, 2004 PHY 184 21 Eddy currents > > Eddy currents often undesirable since they dissipate energy in form of heat Moving conducting parts often laminated o o > Build up several thin layers separated by nonconducting material Layered structure confines eddy currents to individual layers Used in transformers and motors to minimize eddy currents and improve efficiency March 16, 2004 PHY 184 22 Inductance (units) > > > > Inductor is a device used to produce and store a desired B field (e.g. solenoid) A current, i, in an inductor with N turns produces a magnetic flux, ΦB, in its central region N Φ B Inductance, L is defined as L = i SI unit is henry, H 2 H = T ⋅m / A March 16, 2004 PHY 184 23 Inductance of a solenoid NΦ B L= i > What is inductance of a solenoid? > First find flux of single loop in solenoid > # of turns (N ) per unit length (l ) r r ΦB = ∫ B •dA = BA = µ0 niA 2 L = lµ 0 n A n = N /l L 2 = µ0n A or l > Thus > Depends only on the physical properties of the solenoid March 16, 2004 PHY 184 24