Physics 169 Kitt Peak National Observatory Luis anchordoqui 9.19.1 Inductance Inductance An inductor stores energy in magnetic field just a capacitor in electric An inductor stores energy in theasmagnetic fieldstores just asenergy a capacitor stores field energ n the electric field. AWe changing B-field willthat leada to an induced a circuit have shown earlier changing B-fieldemf willinlead to an induced emf in circuit. Question Question : If generates a circuit generates a changing field, does it lead to an If a circuit a changing magneticmagnetic field it lead to an induced emf in same circuit? induced emf in the samedoes circuit? YES! Self-Inductance YES! Self-Inductance The inductance L of any current element is Inductance L of any current element is di The negative sign di EL = VL = L Negative sign from from LenzLenz LawLaw. E L = VL = L dt comes comes dt VS Vs Vs Unit of L: Henry(H) 1H=1· Unit of L : Henry (H) 1H = 1 1· H = A1 A A • •All (including have some inductance. Allcircuit circuitelements elements (includingresistors) resistors) have some inductance Commonlyused usedinductors: inductors:solenoids, solenoidstoroids and toroids • •Commonly circuitsymbol: symbol • •circuit Vs 1H=1 A Example : Solenoid Example Solenoid EL = V B VA = L VB < VA di <0 dt EL = V B VA = VB > VA L di >0 dt Recall Faraday’s Law EL = where N d B dt B is magnetic flux ☛ = N d (N dt B) B is flux linkage ) Alternative definition of Inductance d di N (N B ) = L ) L = dt dt i ) Inductance is also flux linkage per unit current B (1) Solenoid: To first order approximation, Calculating Inductance: ① Solenoid B = µ0 ni where n = N/L = Number of coils per unit length. To first order approximation B = µ0 ni Consider a subsection of length l of the solenoid: n = N/` ☛ number of coils per unit length Flux linkage = N B = nl · BA where A is cross-sectional area Consider a subsection of length l of solenoid Flux linkage = N L= B N B = µ0 n2 lA i L = µ0 n2 A where = Inductance per is unitcross-sectional length l = nl · B A Notice : ) Note ❑L / n2 N A area 2 (i) L ⇤ n2 B 0 (ii) The inductance, like the capacitance, depends only on geometric factors, not on i. L = i = µ n lA L = µ0 n2 A = Inductance per unit length l ❑Inductance (like capacitance) depends only on geometric factors (not on) i I We see that L depends only on the geometrical factors ( n , R and l ) and is independent of the current I . ② Toroid Inside toroid µ0 iN B = 2⇡r Example 11.3 Self-Inductance a Toroidare Recall ☛ B-fieldoflines concentric circles Outside toroid B = 0 9.1. INDUCTANCE Flux linkage through Ztoroid (2) Toroid: N B = N ~ · d~a B 2 n~ (a) B k d~a 109 + Figure 11.2.3 A toroid with N turns da = h dr Recall: B-field lines are concentric circles. Insidebthe toroid: Z (b) µ0 iN h dr µ iN = B= 2⇥r 2⇡ r a (NOT a constant) ⌘ µ0 iN 2 hwhere r ⇣ b is the distance from center. = ln 2⇡ Outside theatoroid: B =20 ⇣b⌘ N B µ0 N h = ln ) Inductance! L = i 2⇡ a 2 Flux linkage through the toroid Again L / N KEY 11-7 0 ˆ ⇤ ⇤ ⌅ 4. In the RL circuit show in Figure 11.14.1, can the self-induced than the emf supplied by the battery? 9. 2 LR Circuits (A) Charging an inductor When switch is adjusted to position a By loop rule (clockwise) E0 VR + VL = 0 a Figure 11.14.1 E0 ) iR di L = 0 dt di R E0 + i = dt L L First Order Differential Equation Similar to equation for charging a capacitor! changing variables x = (E0 /R) dx = i L dx x+ =0 R dt Z x xo 0 dx = 0 x ln(x/x0 ) = R L Z t dt 0 Rt/L x = x0 e Rt/L i = 0 @ t = 0 ) x0 = E0 /R E0 R E0 i= e R Rt/L di di R E0 First Order Differ+ i= ential Equation dt L E0 L t/⌧L Solution ☛ i(t) = (1 e ) Similar to the equation for charging a capacitor! (Chap5) R ⇥ E0 Solution: i(t) = 1 e t/ L ⌧L = L/R ☛ R Inductive time constant where ⇥L = Inductive time constant = L/R | VR | | VL | = iR | VR | = = E0 (1 e t/⌧L t/ ) iR = E0 (1 e ) di di E 0 E0 1 1 t/ L L = =| VLL| = L= L= · L · · · · ·ee t/⌧ =E0 eE0t/e L t/⌧L ) dt dt R R⌧L⇥L L (B) Discharging inductor ”Discharging” an an inductor When switch is adjusted position b afterinductor the inductor has charged been Whenthe switch is adjusted at at position has been b after ”charged” (i.e. current i = E0 /R is flowing in the circuit.). i.e. current i = E0 /R is flowing in circuit Byloop looprule: rule By VLVL ⇤ L di di L dt VRVR= 0= ⇤ iR 0 = 0 iR = dt inductor as source of emf) (Treat Treat inductor as source of di didt + ) + dt 0 emf R Discharging a capacitor Ri = 0 (Chap5) L i = 0 Discharging an inductor L i(t) t/⌧ = i0 e i(t) = i0 e t/ L L where i0 = i(t = 0) = Current when the circuit just switch to position b. where i0 = i(t = 0) = Current when circuit just switch to position b + i=0 (Chap5) where i0 = i(t = 0) the circuit just switch to position b. dt = LCurrent when i(t) = i0 e t/ L where i0 = i(t = 0) = Current when the circuit just switch to position b. Summary Summary : During charging of inductor, During charging of inductor 1. At t = 0, inductor acts like open circuit when current flowing is zero. 1. At t = 0 inductor acts like open circuit when current flowing is zero 2. At charging t ⇥ ⌅, ofinductor Summary : During inductor,acts like short circuit when current flowing is stablized at maximum. 2. At t !1. 1 inductor actslikelike circuit flowing At t = 0, inductor acts open short circuit when current when flowing iscurrent zero. 2. At t ⇥ ⌅, inductor acts like short circuit when currentisflowing is stabilized at maximum stablized at maximum. 3. Inductors are used everyday in switches for safety concerns. 3. Inductors are used everyday in switches for safety concerns. 3. Inductors are used everyday in switches for safety concerns Summary 11.6.1 Rising Current (a) I ! = I 2 R + LI dI . dt (b) (11.6.6) The11.6.1 left-hand side represents therule ratefor at inductors which the(a) battery delivers energy to the Figure Modified Kirchhoff’s with increasing current, and circuit. (b) On the other hand, the first term on the right-hand side is the power dissipated in the with decreasing current. See Section 11.4.2 for cautions about use of this modified rule.resistor in the form of heat, and the second term is the rate at which energy is stored in the inductor. While the energy dissipated through the resistor is irrecoverable, the stored in the inductor can be released later. The polarity of the selfThemagnetic modifiedenergy rule for inductors may be obtained as follows: induced emf is such as to oppose the change in current, in accord with Lenz’s law. If the rate of change of current is positive, as shown in Figure 11.6.1(a), the self-induced emf 11.6.2 Decaying up an inducedCurrent current I ind moving in the opposite direction of the current I to ! L sets oppose such an increase. The inductor could be replaced by an emf Next we consider the RL circuit shown in Figure 11.6.5. Suppose the switch S has been | ! L | = L | dI / dt | = + L(dI / dt) with the polarity shown in Figure 11.6.1(a). On the1 other closed for circuit a long timewith so thatrising the current is at its equilibrium value ! / R . circuit What happens RL current and equivalent I hand, if , as shown in Figure 11.6.1(b), the induced current set by modified the dI / dt < 0 Figureto11.6.2 (a) RL Circuit rising Scurrent. (b) and Equivalent circuitindusingupthe 0 switches the current when at t = with is opened S2 closed? 1 Kirchhoff’s loop self-induced emfrule. ! L flows in the same direction as I to oppose such a decrease. RL circuit Consider the whether shown in Figure 11.6.2. At t = 0 the We find We see that the rate of change of current in increasing ( dI /switch decreasing dt > 0 )isorclosed. a .toThis b along bothnot cases, change in potential when moving from theto the < 0 ), indoes that( dIthe/ dtcurrent risethe immediately to its maximum value is due !/R presence ofofthe inductor. the have modified Kirchhoff’s rule for direction theself-induced current I is Vemf Thus, we ! Vina =the ! L(d I / d t) .Using b increasing current, dI / dt > 0 , the RL circuit is described by the following differential equation: Kirchhoff's Loop Rule Modified for Inductors (Misleading, see Section 11.4.2): dI ! "the IR"direction | ! L | = ! of " IR = 0 the . “potential change” (11.6.1) If an inductor is traversed in the" L current, is dt ! L(dI / dt) . On the other hand, if the inductor is traversed in the direction opposite of the RLtheFigure circuit with decaying andand equivalent circuit current, “potential change” iscircuit . + L(dIwith / dt)current (a) RL decaying current, (b) equivalent Note that there is11.6.5 an important distinction between an inductor and a circuit. resistor. The 9.3 Energy Stored in Inductors Inductors stored magnetic energy through magnetic field stored in circuit Recall equation for charging inductors E0 Multiply both sides by i E0 i |{z} Power input by emf (Energy supplied one charge = qE0 ) iR = 2 i R |{z} di L = 0 dt + Joule’s heating (Power dissipated by resistor) di Li | {zdt} Power stored in inductor ) Power stored in inductor Integrating both sides and use initial condition At t = 0, i(t = 0) = UB (t = 0) = 0 ) Energy stored in inductor ☛ UB = 1 Li2 2 Energy Density Stored in Inductors Consider an infinitely long solenoid of cross-sectional area A For a portion l of solenoid L = µ0 n2 lA ) Energy stored in inductor: 1 2 1 UB = Li = µ0 n2 i2 |{z} lA 2 2 Volume of solenoid ) Energy density (= Energy stored per unit volume) inside inductor UB 1 uB = = µ 0 n2 i 2 lA 2 Recall magnetic field inside solenoid B = µ0 ni ) uB B2 = 2µ0 This is a general result of energy stored in a magnetic field Inductance and Magnetic En 9.4 Mutual Inductance Very often the magnetic flux through the area enclosed by a circuit varies with time because of time-varying currents in nearby circuits 11.1 Mutual Inductance Suppose two coils are placed near each other, as shown in Figur mutual inductance depends on interaction of two circuits Consider two closely wound coils of wire shown in cross-sectional view Current I1in coil 1 which has N1 turns creates a magnetic field Some magnetic field lines pass through coil 2 which has N2 turns Figure 11.1.11 Changing current in coil 1coil produces The magnetic flux caused by the current in coil and passing through 2 is changing 12 We define the mutual inductance ofThe coilfirst 2 with respect toand coilcarries 1 coil has a current I1 which give N1 turns ! B1 . The Nsecond coil has N 2 turns. Because the two coils are cl 2 12 M12the⌘magnetic field lines through coil 1 will also pass through I1 through one turn of coil 2 due to I . Now, by v magnetic flux 1 We shall see that the mutual inductance M12 depends only on th of the two coils such as the number of turns and the radii of the t If current I1 varies with time In a ssimilar manner, there a current we see from Faraday’ law that emfsuppose inducedinstead by coil 1 iniscoil 2 is I 2 in varying with time (Figure 11.1.2). Then the induced emf in coil 1 E2 = N2 d 12 dt = d N2 dt ✓ ◆! = " N d# = " d B! M12 I1 dt dI1dt %% = M12 N2 dt 21 21 1 ! $ dA1 , 2 coil 1 and a current is induced in coil 1. If current I2 varies with time ☛ emf induced by coil 2 in coil 1 is E1 = dI2 M21 dt In mutual induction emf induced in one coil 11.1.2 Changing current coil 2 coil produces changing m is always proportional toFigure rate at which current in in other is changing It is easily seen that flux in coil 1 is proportional to the changing curre M12 =This Mchanging = M 21 t = 0 b is suddenly thrown to at . 9. 5 LC Circuit (Electromagnetic Oscillator) ies: Figure 11.13.6 LC circuit After the capacitor is charged we move the switch to position b 9.4 LC Circuit (Electromagnetic Oscillator) Initial charge on capacitor = Q Initial charge on capacitor = Q current== 00 InitialInitial current No battery. No battery current i tobe be in in the direction that that increases charge charge on the positive AssumeAssume current direction decreases i to capacitor plate. on positive capacitor plate dQi = dQ dt ) i = dt of the inductor. By Lenz Law, we also know the ”poles” ⇤ (9.1) (10.1) By Lenz Law we also know poles of inductor Loop rule: Loop rule ☛ V C + VL = VC + VL = 0 Q di L 0C dt = 0 (9.2) Combining equations (9.1) and (9.2), we get Q C di2 L d Q=+ 01 Q = 0 dtdt2 LC Combining equations (10.1) and (10.2) we get This is similar to the equation of motion 2 of a simple harmonic oscillator: d Q 1 + Q= 0 2 dt LC d2x k 2 + x=0 (10.2) This is similar to equation of motion of ilar to theharmonic equationoscillator of motion a simple harmonic oscillator: d2 x k + x= 0 2 dt m Another approach (conservation of energy) 2 dx k + x=0 Total energy stored in circuit 2 dt m = UE U pproach (conservation of energy) gy stored in circuit: U = + UB Q2 1 2 + Li 2C 2 U =is zero UE no + energy UB is dissipated in circuit Since resistance in circuit ) Energy contained in circuit ⇥is conserved ⇥ 2 Q 1 2 dU ) U= = 0 2C + 2 Li dt ⌘ the circu dQis zero, di dQin esistance in the Q circuit no energy is⇣ dissipated ) · + Li = 0 * i = y contained in the C circuit dt is conserved. dt dt ) ) di Q L + = 0 dt C d2 Q 1 + Q = 0 2 dt LC Solution to this differential equation is in form Q(t) = Q0 cos(!t + ) dQ ) = !Q0 sin(!t + ) dt d2 Q 2 ) = ! Q0 cos(!t + ) 2 dt = !2 Q d2 Q 2 ) + ! Q = 0 2 dt 1 2 Angular frequency of LC oscillator ) ! = LC Q0 , dt2 + ⇥2Q = 0 are constants derived from initial conditions 1 Angular frequency 2 dQ ⇥ = (Two initial conditions, e.g. Q(t = 0)LCand i(t are required) of the =LC0)oscillator = dt t=0 2 2 Q the initial Q0 conditions. Also, Q0stored , are constants derived from (Two initial condiEnergy in C = dQ = cos2 (!t + ) tions, e.g. Q(t = 0), and i(t = 0) 2C = dt t=0 are 2Crequired.) 12 Q2 1 2 2Q20 2 2 Energy stored in Energy stored in L C == = cos (⇥t + )Q0 sin (!t + Li = L! 2C 2C 2 ) 2 1 1 Energy stored in L = Li2 = L⇥ 2 Q20 sin2 (⇥t + ) 2 2 1 Q0 2 2 2 * L! = 2 1 = sin + ) Q20 (!t 2 C = ⇥ L⇥ = sin (⇥t + ) 2C C 2C Q20 Q20 Total energy stored = = Total energy stored 2C 2C = Initial energy stored in capacitor ) = Initial energy stored in capacitor Energy oscillations in LC system and mass-spring system LC Circuit Mass-spring System Energy 9. 6 RLC Circuit (Damped Oscillator) In real life circuit ☛ there’s always resistance energy stored in LC oscillator is NOT conserved and 11.8 The RLC Series Circuit We now consider a series RLC circuit that contains a res dU 2 11.8.1. capacitor, as shown in Figure = i R = Power dissipated in resistor dt Negative sign shows that energy U is decreasing i ) ) z}|{ Q dQ di Li + · = dt C dt d2 Q R dQ 1 + · + Q = 0 2 dt L dt LC RLC circuit FigureJoule’s 11.8.1 A series heating i2 R This is similar to equation of motion of a damped harmonic oscillator ~ = b~v ) (e.g. if a mass-spring system faces a frictional force F Solution to equation is of form R 2L t Q(t) = Q0 e| {z } cos(! 0 t + ) | {z } exponential decay term 0 ! = !0 = s s 1 LC !02 ✓ oscillating term ◆2 R 2L ✓ ◆2 R 2L R damping factor = 2L There are three possible scenarios depending on the relative values of and !0 n is less than the undamped oscillation, ! ' < ! 0 . The qualitative Case Underdamping arge on I:the capacitor as a!function of time is shown in Figure 11.10.1. 0 > Figure 11.10.1 Underdamped oscillations ppose the initial condition is Q(t = 0) = Q0 . The phase constant is then Underdamped oscillator always oscillates at a lower frequency than natural frequency of oscillator Case II: Overdamping ts Q1 and Q2 can ! < 0 be determined from the initial conditions Case III:11.10.2 Critical damping !0 = Figure Overdamping and critical damping damping