2 Main PointsInductance and InductorsEnergy in Inductors and in the Magnetic FieldRL and RC CircuitsLC CircuitsRLC Circuits, Damped Oscillations
3 24-1 Inductance and Inductor Faraday’s Law: Changing current in a circuit will induce emf in that circuit as well as others nearbySelf-Inductance: Circuit induces emf in itselfMutual Inductance: Circuit induces emf in second circuit
4 Self InductanceSwitch closesSelf-Induction: changing current through a loop inducing an opposing emf in that same loop.
5 The flux, therefore, is also proportional to the current. The magnetic field produced by the current in the loop shown is proportional to that current:IThe flux, therefore, is also proportional to the current.We define this constant of proportionality between flux and current to be the inductance, L.
6 Combining with Faraday’s Law gives the emf induced by a changing current: Self-induced emfThe minus sign indicates that—as the law states—the self-induced emf e has the orientation such that it opposes the change in current I
7 (a) The current i is increasing and the self-induced e appears along the coil in a direction such that it opposes the increase.(b) The current i is decreasing and the self-induced emf appears in a direction such that it opposes the decrease.The inductance L is a proportionality constant that depends on the geometry of the circuitSI :H (Wb/A)
8 If extra material (e.g., an iron core) is added, the The inductance of an inductor (a set of coils in some geometry; e.g., solenoid) then, can be calculated from its geometry alone if the device is constructed from conductors and air (similar to the capacitance of a capacitor).If extra material (e.g., an iron core) is added, theinductance will increase (just as adding a dielectricincreases capacitance)Archetypal inductor is a long solenoid, just as a pair of parallel plates is the archetypal capacitor.dAr << llrN turns
9 Example: The length and radius of a long, tightly wound solenoid with N turns are l and R respectively. Find its self-inductance?Solution:The magnitude of magnetic field in the solenoid isThe magnetic flux through N turns of wires isThe self-inductance of the long, tightly wound solenoid is
10 Example A coaxial cable is composed of two long, coaxial cylinders with radius R1 and R2 respectively. They carry equal currents I in opposite directions. What is the self-inductance per unite length?Solution:Magnitude of the magnetic field inside and outside the coaxial cable is
11 self-inductance per unite length In series The magnetic flux through an element area dA in a rectangular region as shown in the figure isdrself-inductance per unite lengthIn series
12 The magnetic flux through circuit 2 is Mutual InductionA changing current i1 in circuit 1 causes a changing flux 21 through circuit 2. Then an induced emf appears in circuit 2 .dAThe magnetic flux through circuit 2 isM21 is a constant that depends on the geometry of the two circuits and the magnetic properties of the material.
13 emf induced in circuit 2 by changing currents in circuit 1, through mutual inductance:
14 Example Calculate the mutual inductance for two tightly wound concentric solenoids shown in figure belowSolution:A current I1 in the inner solenoid sets up a magnetic field B1the flux 21 isThe mutual inductance is
15 We can calculate the mutual inductance by assuming that the outer solenoid carries a current I2 The mutual inductances are equalThis is a general result
16 Example: A long, straight wire carrying a current I=I0sint and a rectangular loop whose short edges are parallel to the wire, as shown in the figure. Find the mutual inductance and the mutual induced emf in the rectangular loop.Solution:The magnetic flux through the rectangular loop isThe mutual inductance isThe mutual induced emf in the rectangular is
17 Kirchhoff’s First Rule “Loop Rule” or “Kirchhoff’s Voltage Law (KVL)” "When any closed circuit loop is traversed, the algebraic sum of the changes in potential must equal zero."KVL:
18 Rules for potential differences across various circuit elements.
19 Our convention:We choose a direction for the current and move around the circuit in that direction.When a battery is traversed from the negative terminal to the positive terminal, the voltage increases, and hence the battery voltage enters KVL with a + sign.When moving across a resistor, the voltage drops, and hence enters KVL with a - sign.e1R1R2e2I+ e1- e2- IR1- IR2= 0
20 Modification of Kirchhoff’s loop rule: In moving across an inductor of inductance L along the presumed direction of the current I, the potential change is e = –L dI/dt
21 Example A device as shown below is constructed from conductors Example A device as shown below is constructed from conductors. Find the self-inductance.SolutionSuppose there is a current I in the conductorllIadaIf there is a magnetic field which is parallel to the axis of the cylindrical shell, Find the current
23 24-2 Energy in InductorsLoop ruleAn RL circuitWork done by the batteryEnergy stored in the inductorEnergy lost in a resistor in the form of thermal energy
24 Where is the Energy Stored? An inductor is a device for storing energy in amagnetic field.We can integrate that term to find an expression for UL (starting from zero current)Where is the Energy Stored?
25 24-3 Energy in Magnetic Fields Energy is stored in the magnetic field itself (just as in the capacitor / electric field case).For an ideal solenoidWe can express UL in terms of B(=nI)giving the energy density of the magnetic field:A general result
26 Energy is located within the electric and magnetic fields themselves
27 Example A long coaxial cable consists of two thin-walled concentric conducting cylinders with radii a and b. The inner cylinder carries a steady current I, the outer cylinder providing the return path for that current. The current sets up a magnetic field between the two cylinders. (a) Calculate the energy stored in the magnetic field for a length l of the cable. (b) Calculate the self inductance for a length l of the cable.I
28 Solutionfind H by using Ampere's lawIThe energy densityThe energy
29 Example An air-filled, closely wound toroidal solenoid with inner radius R1 and outer radius R2 , has N turns. Its cross-section is rectangular. Find magnetic-field energy.Solution:By using Ampere’s Law, the magnitude of the magnetic field isThe magnetic energy density
30 The total magnetic energy The volume of a thin cylinder with radius r and thickness dr, as shown in the figure, isThe total magnetic energydrrWhat is the inductance of this toroid?
31 k I R L 24-4 Time Dependence in RL Circuits When the switch K1 is closed, the inductor keeps the current from attaining its maximum value immediately.An RL circuiteLR1k2ICurrent as a function of time:
32 An RL circuiteLR1k2IThe current in an inductor never changes instantaneously, but after the current settles down to a constant value, the inductor plays no role in the circuit
33 LRe1k2IWhen the switch is thrown to K2 the inductor keeps the current from dropping to zero immediately.tItime constantCurrent as a function of time:The time constant determines how fast the current changes with time
34 General rule: inductors resist change in current Hooked to current sourceInitially, the inductor behaves like an open switch.After a long time, the inductor behaves like anideal wire.Disconnected from current sourceInitially, the inductor behaves like a current source.After a long time, the inductor behaves like anopen switch.
36 ACT At t=0 the switch is thrown from position b to position a in the circuit shown: LIWhat is the value of the current I0 immediately after the switch is thrown?(a) I0 = 0(b) I0 = e/2R(c) I0 = 2e/RWhat is the value of the current I¥ a long time after the switch is thrown?(a) I¥ = 0(b) I¥ = e/2R(c) I¥ = 2e/R
37 What is the value of the current I0 just after the switch is thrown? After a long time, the switch is thrown from position a to position b as shown:What is the value of the current I0 just after the switch is thrown?(a) I0 = 0(b) I0 = e/2R(c) I0 = 2e/RRabeLI
38 If the electric field exceeds the “dielectric strength” aIIRIf after a long time, the switch is opened abruptly from position a instead of being thrown to position b, What happens?eLRJust after the switch is thrown, the inductor induces an emf to keep current flowing: emf = L dI/dt (can be much larger than e)However, now there’s no place for the current to go charges build up on switch contacts high voltage across switch gapIf the electric field exceeds the “dielectric strength”(~30 kV/cm in air) breakdown SPARK!
39 ACT1) At time t = 0 the switch is closed. What is the current through the circuit immediately after the switch is closed?a) I = b) I = V/R c) I = V/2RInitially, the inductor acts like an open switch2) What is the current through the circuit a long time after the switch is closed?After a long time, the inductor acts like an ideal wire,a) I = b) I = V/R c) I = V/2R
40 What is the relation between tI and tII? ACT At t=0, the switch is thrown from position b to position a as shown:Let tI be the time for circuit I to reach 1/2 of its asymptotic current.Let tII be the time for circuit II to reach 1/2 of its asymptotic current.What is the relation between tI and tII?eabLIRII(a) tII < tI(b) tII = tI(c) tII > tIWe must determine the time constants of the two circuits by writing down the loop equations.tI=L/2RtII=2L/RThis confirms that inductors in series add
41 Time Dependence in RC Circuits RC circuit with battery and switchI+-Switch at position a: battery charges capacitorLoop rule:
42 Charge as a function of time: Current as a function of time:Conclusion:Capacitor reaches its final charge(Q=Ce ) exponentially with time constant t = RC.Current decays from max (e /R) with same time constant.
43 Suppose that the switch has been in position a for a long time, the capacity is fully charged, and there is no current. We throw the switch to position b: capacitor discharges through resistorI+-
44 Charge as a function of time: Current as a function of time:The negative sign indicates that the actual current is opposite in direction to the current we assumedConclusion:Capacitor discharges exponentially with time constant t = RCCurrent decays from initial max value (-e/R) with same time constant
45 Basic principle: Capacitor resists rapid change in Q resists rapid changes in VCharging (it takes time to put the final charge on)Initially, the capacitor behaves like a wireAfter a long time, the capacitor behaves like anopen switch.DischargingInitially, the capacitor behaves like a battery.After a long time, the capacitor behaves likea wire.
46 What is the value of the current I0+ just after the switch is thrown? ACT At t=0 the switch is thrown from position b to position a in the circuit shown: The capacitor is initially uncharged.What is the value of the current I0+ just after the switch is thrown?abeRCI(a) I0+ = 0(b) I0+ = e /2R(c) I0+ = 2e /RWhat is the value of the current I¥ after a very long time?(a) I¥ = 0(b) I¥ = e /2R(c) I¥ > 2e /R
47 Compare the charge on the two capacitors a short time after t = 0 ACT The two circuits shown below contain identical fully charged capacitors at t=0. Circuit 2 has twice as much resistance as circuit 1.Compare the charge on the two capacitors a short time after t = 0Initially, the charges on the two capacitors are the same. But the two circuits have different time constants:t1 = RC and t2 = 2RC. Since t2 > t1 it takes circuit 2 longer to discharge its capacitor. Therefore, at any given time, the charge on capacitor 2 is bigger than that on capacitor 1.a) Q1 > Q2b) Q1 = Q2c) Q1 < Q2
48 24-5 Oscillations in LC Circuits Set up the circuit above with capacitor, inductor, resistor, and battery.Let the capacitor become fully charged.Throw the switch from a to bWhat happens?
49 Start with a fully charged capacitor the loop rule:Simple harmonic oscillator
50 solution:Q0and f are determined from initial conditions.Here,Q0 (initial condition) determines the amplitude of the oscillationsThe frequency of the oscillations is determined by the circuit parameters (L, C)
51 It undergoes simple harmonic motion, just like a mass on a spring, with trade-off between charge on capacitor (Spring) and current in inductor (Mass)
52 ACT At t=0 the capacitor has charge Q0; the resulting oscillations have frequency w0. The maximum current in the circuit during these oscillations has value I0.What is the relation between w0 and w2, the frequency of oscillations when the initial charge = 2Q0?LC+-Q=t=0(a) w2 = 1/2 w0(b) w2 = w0(c) w2 = 2w0What is the relation between I0 and I2, the maximum current in the circuit when the initial charge = 2Q0?(a) I2 = I0(b) I2 = 2I0(c) I2 = 4I0
59 In a pure LC circuit, energy is transferred back and forth between the capacitor’s electric field and the inductor’s magnetic field.
60 Resistance is Introduced The power dissipated in the resistora resistor causes energy loss, which shows up as heat.The rate of change of the energy in the capacitor and in the inductorEqual to the power loss in the resistor
61 Example A 1. 5 mF capacitor is charged to 57 V Example A 1.5 mF capacitor is charged to 57 V. The charging battery is then disconnected, and a 12 mH coil is connected in series with the capacitor so that LC oscillations occur at time t=0. (a) What is the maximum current in the coil? (b) What is the potential difference VL(t) across the inductor as a function of time? (c) What is the maximum rate (di/dt)max at which the current i changes in the circuit? Assume that the circuit contains no resistance.Solution(a)
63 Example A series RLC circuit has inductance L = 12 mH, capacitance C = 1.6 mF, and resistance R = 1.5 W. (a) At what time t will the amplitude of the charge oscillations in the circuit be 50% of its initial value? (b) How many oscillations are completed within this time?Solution
64 SummaryDefinition of inductance:Induced emf:emf induced in a second loop:Energy in an inductor:Energy density of a magnetic field: