Lecture 23 Chapter 31 Induction and Inductance Review Forces due to B fields r r charge On a moving A current generates a B field FB qv B r On a current r FB iL B Current carrying coil r rfeelsra torque B NiA Biot Savart law r r r 0 id s r dB 3 4 r Ampere s law r r B d s 0 i enc Review Calculated B field for Long straight wire 0i B 2 r At center of loop B 0i 2R Solenoid B 0in Force on a wire carrying current i1 due to B of another parallel wire with current i2 0 Li1i2 F 2 d Force is attractive repulsive if current in both wires are same opposite directions n N L 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 current in loop Current disappears when stops Move magnet away from loop again appears but in direction Faster motion produces a current causes magnet current opposite greater Inductance 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 direction Inductance 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 induction Inductance 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 field Inductance 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 flux analogous to electric flux Inductance 6 Magnetic flux through area A r r dA B B dA dA is vector of magnitude that is to the differential area dA If B is uniform and to A then SI unit is the weber Wb B BA 1Wb 1T m 2 Inductance 7 Faraday s law of induction d B induced emf in loop is equal to the rate at which the magnetic E dt flux changes with time Minus signs means induced emf tends to oppose the flux change If magnetic flux is through a d B closely packed coil of N turns E N dt Inductance 8 Can change the magnetic flux through a loop or coil by If B is constant within coil r r B B dA BA cos Change magnitude of B field within coil Change area of coil or portion of area within field Change angle between B field and area of coil e g rotating coil d B E N dt dB E NAcos dt BE dA NB cos dt d cos E NBA dt Inductance 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 dB dB b then d e tie then E NAcos NA a c zero dt dt 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 B field in same direction to th d Inductance 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 current Inductance 12 Situation a From Lenz s law Bi from induced current opposes increasing B so Bi is into page From right hand rule induced current is clockwise in both sections of circle Do same for situation b and c a b tie then c zero Inductance 13 What is magnitude and direction of induced emf around 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 B 4t x 2 2 Since magnitude B is changing in time flux d B E through the loop is changing so use Faraday s dt law to calculate induced emf Inductance 14 B is not uniform so need to calculate magnetic flux using B r r B dA B to plane of loop and only changes in x direction r r B d A BdA BHdx Treat time as constant so B BHdx 4t H 2 3 0 3 x x dx 4 t H 72 t 2 3 0 3 2 2 Inductance 15 Now use Faraday s law to find the magnitude of the induced emf d B d 72 t E 144 t dt dt 2 At t 0 10s emf 14 V Find direction of emf by Lenz s law B is increasing so Bi is in opposite direction out of the page Right hand rule current and emf are counterclockwise Inductance 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 B BA BLx Inductance 17 Using Faraday s law d B d dx E BLx BL dt dt dt Remember v dx dt so E 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 and current is clockwise Inductance 18 Since loop carries current through a B field there is force given by a r r r FB i L 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 r Fapp Force F1 opposes your force r F1 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 E BLv c d tie then a b tie
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