Lecture 28: MON 23 MARPowerPoint PresentationSlide 3RL Circuit: Fluxing up an InductorSlide 5Fluxing Down an InductorInductors & EnergySlide 8ExampleSlide 10Slide 11Slide 12Lecture 28: MON 23 MARLecture 28: MON 23 MARCh.30.10-11 Inductors & Inductors & InductanceInductancePhysics 2102Jonathan DowlingNikolai TeslaQuickTime™ and a decompressorare needed to see this picture.EXAM 03: 6PM THU 02 APR 2009The exam will cover: Ch.28 (second half) through Ch.32.1-3 (displacement current, and Maxwell's equations).The exam will be based on:HW08 – HW11 Final Day to Drop Course: FRI 27 MAR0Consider a solenoid of length that has loops of area each, and windings per unit length. A current flows through the solenoid and generates a uniform magnetic field inNNA niB niμ==Inductancellside the solenoid.The solenoid magnetic flux is .BNBAΦ =lBr( )20The total number of turns . The result we got for the special case of the solenoid is true for any inductor: . Here is a constant known as the of the solenoiBBN n n A iLi Lμ= → Φ =Φ =l l inductance2200d. The inductance depends on the geometry of the particular inductor. For the solenoid, .Bn AiL n Ai iμμΦ= = =Inductance of the Solenoidll L =μ0n2lA=μ0N / l( )2lA=μ0N2/ lA#RL Circuit: Fluxing up an RL Circuit: Fluxing up an InductorInductor•How does the current in the circuit change with time? −iR + E − Ldidt= 0 i t( )=ER1−e−t/τ( ) i t( )=ER1−e−t/τ( )Time constant of RL circuit: = L/RTime constant of RL circuit: = L/RtE/Ri(t)Fast/Small Slow/Large i(t)RL Circuit Fluxing UPQuickTime™ and aAnimation decompressorare needed to see this picture. i t( )=ER1−e−t/τ( ) i t( )=ER1−e−t/τ( )i 0( )=0i 0( )=0 ′i 0( )=EL ′i 0( )=EL VL0( )=E VL0( )=E ′i t( )=ELe−t /τ ′i t( )=ELe−t /τ VLt( )=L′i t( )=Ee−t/τ VLt( )=L′i t( )=Ee−t/ττ =L / R i ∞( )=ER i ∞( )=ER′i ∞( )= 0′i ∞( )= 0VL∞( )=0VL∞( )=0Fluxing Down an Fluxing Down an InductorInductor0=+dtdiLiR i t( )=ERe−Rt/L=ERe−t/τ i t( )=ERe−Rt/L=ERe−t/τtE/RExponential defluxingi(t)iThe switch is at a for a long time, until the inductor is charged. Then, the switch is closed to b.What is the current in the circuit?Loop rule around the new circuit:Inductors & EnergyInductors & Energy•Recall that capacitors store energy in an electric field•Inductors store energy in a magnetic field. E =iR + Ldidt iE( )= i2R( )+Lididt iE( )= i2R( )+ddtLi22⎛⎝⎜⎞⎠⎟Power delivered by battery= power dissipated by Ri+ (d/dt) energy stored in LPB=BiVB=Bi2RInductors & EnergyInductors & EnergyP =LididtP =LididtUB=Li22UB=Li22Magnetic Potential Energy UB Stored in an Inductor.Magnetic Potential Energy UB Stored in an Inductor.Magnetic Power Returned from Fluxing Down Inductor to Circuit or Put into Fluxing Up Inductor from Circuit.Magnetic Power Returned from Fluxing Down Inductor to Circuit or Put into Fluxing Up Inductor from Circuit.ExampleExample •The switch has been in position “a” for a long time. •It is now moved to position “b” without breaking the circuit.•What is the total energy dissipated by the resistor until the circuit reaches equilibrium?• When switch has been in position “a” for long time, current through inductor = (9V)/(10) = 0.9A.•Energy stored in inductor = (0.5)(10H)(0.9A)2 = 4.05 J•When inductor “defluxes” through the resistor, all this stored energy is dissipated as heat = 4.05 J. 9 V10 10 HlBr0Consider the solenoid of length and loop area that has windings per unit length. The solenoidcarries a current that generates a uniform magnetic field AniB niμ=Energy Density of a Magnetic Fieldl inside the solenoid. The magnetic fieldoutside the solenoid is approximately zero.2 2201The energy stored by the inductor is equal to .2 2This energy is stored in the empty space where the magnetic field is present.We define as energy density where is the volume BBBn A iU LiUu VVμ==l2 2 2 2 2 2 220 0 00 0inside the solenoid. The density .2 2 2 2This result, even though it was derived for the special case of a uniformmagnetic field, holds true in general.Bn A i n i n iBuAμ μ μμ μ= = = =ll20 2BBuμ=Units:uB=[J/m3]Energy Density in E and B FieldsuE=ε0E22uE=ε0E22uB=B22μ0uB=B22μ0The Energy Density of the Earth’s Magnetic Field Protects us from the Solar
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