Lecture 23Chapter 31Induction and InductanceReview• Forces due to Bfields– On a moving charge– On a current• Current carrying coil feels a torque• A current generates a B field–Biot-Savartlaw– Ampere’s lawBvqFBrr×=BLiFBrr×=Brrr×=µτNiA=µencisdB0µ=•∫rr304 rrsidBdrrr×=πµReview• Calculated B field for – Long, straight wire– At center of loop– Solenoid• Force on a wire carrying current, i1, due to B of another parallel wire with current i2• Force is attractive (repulsive) if current in both wires are same (opposite) directionsriBπµ20=RiB20µ=inB0µ=LNn /=diLiFπµ2210=Inductance (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– Current disappears when magnet stops– Move magnet away from loop current again appears but in opposite direction– Faster motion produces a greater currentInductance (2)• Have 2 conducting loops near each other– Close switch so current flows in one loop, briefly register a current in other loop– Open switch, again briefly register current in other loop but in opposite directionInductance (3)• Current produced in the loop is called induced current• The work done per unit charge to produce the current is called an induced emf• Process of producing the current and emf is called inductionInductance (4)• Faraday observed that an induced current (and an induced emf) can be generated in a loop of wire by:– Moving a permanent magnet in or out of the loop– Holding it close to a coil (solenoid) and changing the current in the coil– Keep the current in the coil constant but move the coil relative to the loop– Rotate the loop in a steady B field– Change the shape of the loop in a B fieldInductance (5)• Faraday concluded that an emf and a current can be induced in a loop by changing the amount of magnetic field passing through the loop• Need to calculate the amount of magnetic field through the loop so define magnetic fluxanalogous to electric fluxInductance (6)• Magnetic flux through area A• dA is vector of magnitude dAthat is ⊥⊥⊥⊥ to the differential area, dA• If B is uniform and ⊥⊥⊥⊥ to A then• SI unit is the weber, WbBAB=Φ∫•=Φ AdBBrr211 mTWb ⋅=Inductance (7)• Faraday’s law of induction –induced emf in loop is equal to the rate at which the magnetic flux changes with time• Minus signs means induced emf tends to oppose the flux change • If magnetic flux is through a closely packed coil of N turnsdtdBΦ−=EdtdNBΦ−=EInductance (8)• Can change the magnetic flux through a loop (or coil) by–If B is constant within coil– Change magnitude of Bfield within coil– Change area of coil, or portion of area within Bfield– Change angle between B field and area of coil (e.g. rotating coil) dtdNBΦ−=EθcosBAAdBB=•=Φ∫rrdtdBNAθcos−=EdtdANBθcos−=EdtdNBA)(cosθ−=EInductance (9)• Checkpoint #1 – Graph shows magnitude B(t) of uniform B field passing through loop, ⊥⊥⊥⊥ to plane of the loop. Rank the five regions according to magnitude of emf induced in loop, greatest first. dtdBNAdtdBNA −=−=θcosEb, then d & e tie, then a & c (zero)Inductance (10)• Lenz’s law – An induced emf gives rise to a current whose B field opposes the change in flux that produced it– Magnet moves towards loop the flux in loop increases so induced current sets up B field opposite direction– Magnet moves away from loop the flux decreases so induced current have Bfield in same direction to th dInductance (11)• 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 currentInductance (12)• Situation a –– From Lenz’s law, Bifrom induced current opposes increasing B so Biis into page– From right-hand rule, induced current is clockwise in both sections of circle• Do same for situation b and ca & b tie, then c (zero)Inductance (13)• What is magnitude and direction of induced emfaround loop at t=0.10s? • 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• Since magnitude B is changing in time, flux through the loop is changing so use Faraday’s law to calculate induced emf224 xtB =dtdBΦ−=EInductance (14)• B is not uniform so need to calculate magnetic flux using• B ⊥⊥⊥⊥ to plane of loop and only changes in x direction • Treat time as constant so 23032302272344 txHtdxxHtBHdxB====Φ∫∫∫•=Φ AdBBrrBHdxBdAAdB ==•rrInductance (15)• Now use Faraday’s law to find the magnitude of the induced emf• At t=0.10s, emf = 14 VtdttddtdB144)72(2==Φ=E• Find direction of emf by Lenz’s law– B is increasing so Biis in opposite direction -out of the page– Right-hand rule – current (and emf) are counterclockwiseInductance (16)• If you pull a loop at a constant velocity, v, through a B field, you must apply a constant force, F• As 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 BLxBAB==ΦInductance (17)• Using Faraday’s law • Remember v= dx/dt so• where L is the length of the loop and v is ⊥⊥⊥⊥ to B field• B is decreasing so Biis in same direction (into page) and current is clockwisedtdxBLBLxdtddtdB==Φ=EBLv=EInductance (18)• Since loop carries current through a B field there is a force given by• Use right-hand rule to find direction of FBon segments of loop in B field• Find forces, F2 and F3, cancel each other•Force, F1opposes your forceBLiFBrrr×=1FFapprr−=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. BLv=Ec & d tie, then a & b
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