Chapter 32Inductance L and the stored magnetic energyRL and LC circuitsRLC 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 InductanceThe SI unit for inductance is the henry (H)Named for Joseph Henry: 1797 – 1878American physicistFirst director of the SmithsonianImproved design of electromagnetConstructed one of the first motorsDiscovered self-inductanceAsV1H1⋅=Discussion about Some TerminologyUse emf and current when they are caused by batteries or other sourcesUse induced emf and induced current when they are caused by changing magnetic fieldsWhen 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 CircuitAn 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 increaseDischarging: When S2is connected to position b.PLAYACTIVE FIGURERL Circuit, chargingApplying Kirchhoff’s loop rule to the circuit in the clockwise direction givesSolve for the current I, with initial condition that I(t=0) = 0, we findWhere 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, dischargingWhen 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 givesSolve 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 FieldGiven UL= ½ L I2and assume (for simplicity) a solenoid with L = µon2VSince V is the volume of the solenoid, the magnetic energy density, uBisThis 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 SummaryInductor and capacitor store energy through different mechanismsCharged capacitor Stores energy as electric potential energyWhen current flows through an inductorStores energy as magnetic potential energyA 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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