1P24-Class 24: OutlineHour 1:Inductance & LR CircuitsHour 2:Energy in Inductors2P24-Last Time:Faraday’s LawMutual Inductance3P24-Faraday’s Law of InductionBdNdtεΦ=−Changing magnetic flux induces an EMFLenz: Induction opposes change4P24-Mutual Inductance112 122 112122NMINMIΦ≡Φ→=212dIdtMε≡ −12 21MMM==A current I2in coil 2, induces some magnetic flux Φ12in coil 1. We define the flux in terms of a “mutual inductance” M12:You need AC currents!5P24-Demonstration:Remote Speaker6P24-This Time:Self Inductance7P24-Self InductanceWhat if we forget about coil 2 and ask about putting current into coil 1?There is “self flux”:111 111 NMILINLIΦ≡≡Φ→=dILdtε≡ −8P24-Calculating Self InductanceNLIΦ=Vs1 H = 1 A⋅Unit: Henry1. Assume a current I is flowing in your device2. Calculate the B field due to that I3. Calculate the flux due to that B field4. Calculate the self inductance (divide out I)9P24-Group Problem: SolenoidCalculate the self-inductance L of a solenoid (n turns per meter, length A, radius R)REMEMBER1. Assume a current I is flowing in your device2. Calculate the B field due to that I3. Calculate the flux due to that B field4. Calculate the self inductance (divide out I)LN I=Φ10P24-Inductor BehaviorIdILdtε= −LInductor with constant current does nothing11P24-dILdtε= −Back EMFI0, 0LdIdtε><I0, 0LdIdtε<>12P24-Inductors in CircuitsInductor: Circuit element which exhibits self-inductance Symbol:When traveling in direction of current:dILdtε= −Inductors hate change, like steady stateThey are the opposite of capacitors!13P24-PRS Question:Closing a Switch14P24-LR Circuit0iidIVLdtIRε==−−∑15P24-LR Circuit0 dI L dILIdt R dt RIRεε⎛⎞=⇒ =−−⎜⎟⎝⎠−−Solution to this equation when switch is closed at t = 0:()/() 1tIt eRτε−=−:LRtimeconstantLRτ=16P24-LR Circuitt=0+: Current is trying to change. Inductor works as hard as it needs to to stop itt=∞: Current is steady. Inductor does nothing.17P24-LR CircuitReadings on VoltmeterInductor (a to b)Resistor (c to a)ct=0+: Current is trying to change. Inductor works as hard as it needs to to stop itt=∞: Current is steady. Inductor does nothing.18P24-General Comment: LR/RCAll Quantities Either:()/FinalValue( ) Value 1tteτ−=−/0Value( ) Valuetteτ−=τ can be obtained from differential equation (prefactor on d/dt) e.g. τ = L/R or τ = RC19P24-Group Problem: LR Circuit1. What direction does the current flow just after turning off the battery (at t=0+)? At t=∞?2. Write a differential equation for the circuit3. Solve and plot I vs. t and voltmeters vs. t20P24-PRS Questions:LR Circuit & Problem…21P24-Non-Conservative FieldsR=100ΩR=10ΩBdddtΦ⋅=−∫EsGGI=1AE is no longer a conservative field –Potential now meaningless22P24-This concept (& next 3 slides) are complicated.Bare with me and try not to get confused23P24-Kirchhoff’s Modified 2nd RuleBiidVdNdtΦ∆=− ⋅ =+∑∫EsGGv0BiidVNdtΦ⇒∆− =∑If all inductance is ‘localized’ in inductors then our problems go away – we just have:0iidIVLdt∆−=∑24P24-Ideal Inductor• BUT, EMF generated in an inductor is not a voltage drop across the inductor!dILdtε=−inductor0Vd∆≡−⋅=∫EsGGBecause resistance is 0, E must be 0!25P24-Conclusion:Be mindful of physicsDon’t think too hard doing it26P24-Demos:Breaking circuits with inductors27P24-Internal Combustion EngineSee figure 1:http://auto.howstuffworks.com/engine3.htm28P24-Ignition SystemThe Distributor:http://auto.howstuffworks.com/ignition-system4.htm(A) High Voltage Lead(B) Cap/Rotor Contact(C) Distributor Cap(D) To Spark Plug(A) Coil connection(B) Breaker Points(D) Cam Follower(E) Distributor Cam29P24-Modern IgnitionSee figure:http://auto.howstuffworks.com/ignition-system.htm30P24-Energy in Inductor31P24-Energy Stored in InductordIIR Ldtε=+ +2dIIIR LIdtε=+()2212dIIR LIdtε=+BatterySuppliesResistorDissipatesInductorStores32P24-Energy Stored in Inductor212LULI=But where is energy stored?33P24-Example: Solenoid Ideal solenoid, length l, radius R, n turns/length, current I:0BnIµ=22oLnRlµπ=()22221122BoULI nRlIµπ==222BoBURlπµ⎛⎞=⎜⎟⎝⎠EnergyDensityVolume34P24-Energy Density Energy is stored in the magnetic field!22BoBuµ=: Magnetic Energy Density 22oEEuε=: Electric Energy Density35P24-Group Problem: Coaxial CableXIIInner wire: r=aOuter wire: r=b1. How much energy is stored per unit length? 2. What is inductance per unit length?HINTS: This does require an integralThe EASIEST way to do (2) is to use (1)36P24-Back to Back EMF37P24-PRS Question:Stopping a
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