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SMU PHYS 1304 - Lecture Notes

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Chapter 32Inductance L and the stored magnetic energyRL and LC circuitsRLC circuitResistance, Capacitance and InductanceOhm’s Law defines resistance:VRI∆≡Capacitance, the ability to hold charge:QCV∆≡Capacitors store electric energy once charged:Resistors do not store energy, instead they transform electrical energy into thermo energy at a rate of:22VP V I I RR∆∆= ⋅ = =( )221 12 2EQU C VC∆ ∆= =Inductance, the ability to “hold” current (moving charge).Inductors store magnetic energy once “charged” with current, i.e., current flows through it.Inductance, the definitionIWhen a current flows through a coil, there is magnetic field established. If we take the solenoid assumption for the coil:EEL+–0B nIµ=When this magnetic field flux changes, it induces an emf, EL, called self-induction:()()020BLd NAB d NA nIddI dIn V Ldt dt dt dt dtµΦµ= − = − = − = − ≡ −Eor:LdILdt≡ −EThis defines the inductance L, which is constant related only to the coil. The self-induced emf is generated by current flowing though a coil. According to Lenz Law, the emf generated inside this coil is always opposing the change of thecurrent which is delivered by the original emf.For a solenoid:20L n Vµ=Wheren: # of turns per unit length.N: # of turns in length l.A: cross section areaV: Volume for length l.InductorWe used a coil and the solenoid assumption to introduce the inductance. But the definition holds for all types of inductance, including a straight wire. Any conductor has capacitance and inductance. But as in the capacitor case, an inductor is a device made to have a sizable inductance. LLdIdt≡ −EAn inductor is made of a coil. The symbol is Once the coil is made, its inductance L is defined. The self-induced emf over this inductor under a changing current I is given by: LdILdt= −EUnit for InductanceThe SI unit for inductance is the henry (H)Named for Joseph Henry: 1797 – 1878American physicistFirst director of the SmithsonianImproved design of electromagnetConstructed one of the first motorsDiscovered self-inductanceAsV1H1⋅=Discussion about Some TerminologyUse emf and current when they are caused by batteries or other sourcesUse induced emf and induced current when they are caused by changing magnetic fieldsWhen dealing with problems in electromagnetism, it is important to distinguish between the two situationsExample: Inductance of a coaxial cable 2ln2boBaoµ IB dA drπrµ Ibπ aΦ = = =  ∫ ∫ln2oBµbLIπ aΦ = =  BLddILdt dtΦ= − = −EStart from the definitionWe have, or B Bd LdI LIΦ Φ= =So the inductance isPut inductor L to use: the RL CircuitAn RL circuit contains a resistor R and an inductor L. There are two cases as in the RC circuit: charging and discharging. The difference is that here one charges with current, not charge.Charging:When S2is connected to position a and when switch S1is closed (at time t = 0), the current begins to increaseDischarging: When S2is connected to position b.PLAYACTIVE FIGURERL Circuit, chargingApplying Kirchhoff’s loop rule to the circuit in the clockwise direction givesSolve for the current I, with initial condition that I(t=0) = 0, we findWhere the time constant is defined as:0d Iε I R Ldt− − =( ) ( )τ− −= − ≡ −1 1Rt L tε εI e eR RHere because the current is increasing, the induced emf has a direction that should oppose this increase.τ≡LRRL Circuit, dischargingWhen switch S2is moved to position b, the original current disappears. The self-induced emf will try to prevent that change, and this determines the emf direction (Lenz Law).Applying Kirchhoff’s loop rule to the previous circuit in the clockwise direction givesSolve for the current I, with initial condition that we find− + =0d II R Ldtτ− −= ≡Rt L tε εI e eR R()= =E0RI tEnergy stored in an inductorIn the charging case, the current I from the battery supplies power not only to the resistor, but also to the inductor. From Kirchhoff’s loop rule, we have= +d Iε I R LdtMultiply both sides with I:= +2d IεI I R LIdtThis equation reads: powerbattery=powerR+powerLSo we have the energy increase in the inductor as:=LdUd ILIdt dtSolve for UL:= =∫2012ILU LId I LIStored energy type and the Energy Density of a Magnetic FieldGiven UL= ½ L I2and assume (for simplicity) a solenoid with L = µon2VSince V is the volume of the solenoid, the magnetic energy density, uBisThis applies to any region in which a magnetic field exists (notjust the solenoid) = =  22212 2L oo oB BUµ n V Vµ n µ≡ =22LBoUBuVµSo the energy stored in the solenoid volume V is magnetic (B) energy.And the energy density is proportional to B2.RL and RC circuits comparisonEnergyDischargingChargingRCRL−=Rt LεI eR( )−= −1Rt LεI eR=212LU LI221( )2 2CQU C VC∆= =( )−=tRCεI t eR( )−=tRCQI t eRCEnergy densityElectric fieldMagnetic field=22BoBuµ=212E ouε EEnergy Storage SummaryInductor and capacitor store energy through different mechanismsCharged capacitor Stores energy as electric potential energyWhen current flows through an inductorStores energy as magnetic potential energyA resistor does not store energy Energy delivered is transformed into thermo energyLC CircuitsLC: circuit with an inductor and a capacitor.Initial condition: either the C or the L has energy stored in it. The “show” starts: when the switch S closes, t = 0 and the time starts.Your physics intuition: neither Cnor L consumes energy, the initially stored energy will oscillate between the C and the L.LC Circuits, the calculationInitial condition: Assume that the capacitor was initially charged to Qmax. when the switch S closes, t = 0 and the time starts.0C Lq dIV LC dt∆= − =+ EHere q is the charge in the capacitor at time t. Because charges flow out of the capacitor to form the current I, we have:dqIdt−=Apply Kirchhoff’s loop rule:Combine these two equations:2210d IIdt LC+ =Solve for the current I: ()maxI I sin tω=21, and max maxI QLCω ω≡ =withHere we also have ()maxq Q cos tω=LC Circuits, the oscillation of charge and currentOscillations: simply plot the results, we find out that the charge stored in the capacitor and the current “stored” in the inductor oscillate. The phase difference is T/2.This means that when the capacitor is fully charged, the current is zero. When the


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SMU PHYS 1304 - Lecture Notes

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