Lecture 25 Chapter 31 Induction and Inductance Magnetic flux Review r r B B dA Faraday s law Changing B field produces an E field Restate Faraday s law d B r r d B E E ds dt dt Lenz s law induced Inductance L defined emf gives rise to a current whose B field N B opposes the change in L flux that produced it i Review Inductor device produces known B field Solenoid is an inductor with inductance per unit length of L 2 0n A l Self induce emf EL appears in any coil in which the current is changing di EL L dt Direction of EL follows Lenz s law and opposes the change in current Review RL circuit resistor and inductor in series Time dependence on current in RL circuit Initially inductor acts to oppose changes in current through it Long time later inductor acts like simple wire Rise of current E i 1 e t L R Decay of current E t L i e i0e t L R Inductive time constant L L R Inductance 39 Mutual induction current in one coil induces emf in other coil Distinguish from selfinduction Mutual inductance M21 of coil 2 with respect to coil 1 is M 21 N 2 21 i1 Inductance 40 M 21 N 2 21 i1 Rearrange equation M 21i1 N 2 21 Vary i1 with time d 21 di1 N2 M 21 dt dt Faraday s law d 21 E2 N 2 dt Induced emf in coil 2 due to i in coil 1 is di1 E2 M 21 dt Obeys Lenz s law minus sign Inductance 41 Reverse roles of coils What is induced emf in coil 1 from a changing current in coil 2 Same game as before M 12 N 1 12 i2 di2 E1 M 12 dt Inductance 42 The mutual inductance terms are equal M 21 M 12 M Rewrite emfs as di1 E2 M dt di2 E1 M dt Notice same form as self induced emf di EL L dt N B L i Inductance 43 Generators convert mechanical energy to electrical energy External agent rotates loop of wire in B field Hydroelectric plant Coal burning plant Changing B induces an emf and current in an external circuit Inductance 44 Alternating current ac generator Ends of wire loop are attached to slip rings which rotate with loop Stationary metal brushes are in contact with slip rings and connected to external circuit emf and current in circuit alternate in time E t Inductance 45 Calculate emf for generator with N turns of area A and rotating with constant angular velocity Magnetic flux is r r B B dA BA cos Relate angular displacement to angular velocity t Flux through one loop is B BA cos t Inductance 46 Faraday s law says d B E N dt Substitute B BA cos t Maximum emf is when t 90 or 270 degrees E max NBA Emf is 0 when t 0 or 180 degrees d E NBA cos t dt E NBA sin t E t Inductance 47 Direct current dc generator E Ends of loop are connected to a single split ring t Metal brush contacts to Not suitable for most split ring reverse their applications roles every half cycle Can use to charge Polarity of induced emf batteries reverses but polarity of split ring remains the Commercial dc gen same use out of phase coils Inductance 48 Motors converts electrical energy to mechanical energy Generator run in reverse Current is supplied to loop and the torque acting on the current carrying loop causes it to rotate Do mechanical work by using the rotating armature As loop rotates changing B field induces an emf Induced emf back emf reduces the current in the loop remember Lenz s law Power requirements are greater for starting a motor and for running it under heavy loads Inductance 49 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 Inductance 50 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 Inductance 51 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 Direction of eddy currents produce a drag force on the moving vehicle Eddy currents decrease steadily as car slows giving a smooth stop Inductance 52 Eddy currents often undesirable since they dissipate energy in form of heat Moving conducting parts often laminated 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 Inductance 53 How much energy is stored in a B field Conservation of energy expressed in loop rule di E L iR dt Multiply each side by i di Ei Li i2R dt Ei is rate at which emf device delivers energy to rest of circuit i 2R is rate at which energy appears as thermal energy in resistor Inductance 54 Middle term represents Integrating gives the rate dUB dt at UB i which energy is stored dU B Lidi in the B field 0 0 di 2 Ei Li i R dt Energy stored in B field U dU B di Li dt dt dU B Lidi B 1 Li 2 Similar to UE U E 1 q2 2 C 2 Inductance 55 What is the energy density of B field Energy density uB is energy per unit volume Volume is area x length UB uB Al U B 1 Li 2 Substituting UB gives Li 2 uB l2 A For a solenoid L 0n 2 A l Energy density is 2 1 2 2 uB 0n i 2 Inductance 56 Substituting B gives magnetic energy density 1 2 2 uB 0n i 2 Remember B field from a solenoid is B o in 2 1 B uB 2 0 Similar to electric energy density 1 2 uE 0 E 2 Inductance 57 Place coil C at center of long solenoid which has a steadily decreasing current What is the magnitude of the induced emf in coil C Solenoid generates uniform B field of B o in Current is decreasing so B field decreases Inductance 58 Since B field decreases the flux decreases and an emf is induced in coil C Faraday s law r r B B dA E N d B dt The current is decreasing at a steady rate so flux also decreases at steady rate and write d B B B f B i dt t t Inductance 59 Need to …
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