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Physics 122B Electricity and Magnetism Lecture 23 (Knight: 33.8-33.10) Inductance and LR Circuits Martin Savage Lecture 23 Announcements Lecture HW has been posted and is due on Wednesday at 10 PM. Midterm examination 3 is this coming Friday--questions from lecture, tutorial AND Lab. 5/25/2017 Physics 122B - Lecture 23 2 Generators The figure shows a coil with N turns rotating in a magnetic field, with the coil connected to an external circuit by slip rings that transmit current independent of rotation. The flux through the coil is: m A B AB cos AB cos t Ecoil N d m d ABN cos t ABN sin t dt dt Therefore, the device produces emf and current that will vary sinusoidally, alternately positive and negative. This is called an alternating current generator, producing what we call AC voltage. 5/25/2017 Physics 122B - Lecture 23 3 Example: An AC Generator A coil with area 2.0 m2 rotates in a 0.10 T magnetic field at a frequency of 60 Hz. How many turns are needed to generate an AC emf with a peak voltage of 160 V? Ecoil ABN sin t Emax ABN 2 f Emax (160 V) N 21 turns 2 2 f AB 2 (60 Hz)(2.0 m )(0.10 T) 5/25/2017 Physics 122B - Lecture 23 4 Transformers When a coil wound around an iron core is driven by an AC voltage V1cos wt, it produces an oscillating magnetic field that will induce an emf V2cos wt in a secondary coil wound on the same core. This is called a transformer. The input emf V1 induces a current I1 in the primary coil that is proportional to 1/N1. The flux in the iron is proportional to this, and it induces an emf V2 in the secondary coil that is proportional to N2. Therefore, V2 = V1(N2/N1). From conservation of energy, assuming no losses in the core, V1I1 = V2I2. Therefore, the currents in the primary and secondary are related by the relation I1 = I2(N2/N1). A transformer with N2>>N1 is called a step-up transformer, which boosts the secondary voltage. A transformer with N2<<N1 is called a stepdown transformer, and it drops the secondary voltage. 5/25/2017 Physics 122B - Lecture 23 5 The Tesla Coil A special case of a step-up transformer is the Tesla coil. It uses no magnetic material, but has a very high N2/N1 ratio and uses high-frequency electrical current to induce very high voltages and very high frequencies in the secondary. There is a phenomenon called “the skin effect” that causes high frequency AC currents to reside mainly on the outer surfaces of conductors. Because of the skin effect, one does not feel (much) the electrical discharges from a Tesla coil. 5/25/2017 Physics 122B - Lecture 23 6 Metal Detectors Metal detectors like those used at airports can detect any metal objects, not just magnetic materials like iron. They operate by induced currents. A transmitter coil sends high frequency alternating currents that will induce current flow in conductors in its field. Because of Lenz’s Law, the induced current opposes the field from the transmitter, so that net field is reduced. A receiver coil detects the reduction in the magnetic fields from the transmitter and registers the presence of metal. 5/25/2017 Physics 122B - Lecture 23 7 (Self-) Inductance We define the inductance L of a coil of wire producing flux m as: m L I The unit of inductance is the henry: 1 henry = 1 H = 1 T m2/A = 1 Wb/A The circuit diagram symbol used to represent inductance is: Example: The inductance of a long solenoid with N turns of cross sectional area A and length l is: 0 NI per turn BA B m N per turn NBA 5/25/2017 0 N A l 2 l I Physics 122B - Lecture 23 Lsolenoid m 0 N 2 A I l 8 Example: Length of an Inductor An inductor is made by tightly winding 0.30 mm diameter wire around a 4.0 mm diameter cylinder. What length cylinder has an inductance of 10 mH? L 0 N 2 A l 0 (l / d )2 ( r 2 ) l 0 r 2l d 2 1.0 105 H d 2L (3.0 104 m)2 (1.0 105 H) l 0.057 m 5.7 cm 2 7 3 2 0 r (4 10 Tm/A) (2.0 10 m) 5/25/2017 Physics 122B - Lecture 23 9 Potential Across an Inductor Ecoil N d per turn dt Ecoil L 5/25/2017 d m dt dI dt Physics 122B - Lecture 23 10 Potential Across an Inductor (2) Ecoil L 5/25/2017 dI dt VL L Physics 122B - Lecture 23 dI dt 11 Inductors and Sparks An inductor responds to an interruption in current flow by developing a very large potential (L dI/dt) across its terminals. This potential is typically dropped across the object (usually a switch) that is effecting the change in current. Therefore, breaking a closed circuit with an inductor often results in a spark across the switch contacts. Over time, this can erode and destroy the switch. Spark plugs in automobiles are driven by inductances that are electronically interrupted. 5/25/2017 Physics 122B - Lecture 23 12 Example: Large Voltage across an Inductor A 1.0 A current passes through a 10 mH inductor coil. What potential difference is induced across the coil if the current drops to zero in 5 s? dI I A)) ((-10. 1.0 A -5 2.0 10 A/s -6 dt t (5.0 10 s) VL L 5/25/2017 dI (0.010 H)(2.0 10-5 A/s) 2000 V dt Physics 122B - Lecture 23 13 Question The potential at a is higher than the potential at b. Which of the following statements about the inductor current I is consistent with this observation? a. b. c. d. e. 5/25/2017 I is from a to b and steady. I is from a to b and increasing I is from a to b and decreasing I is from b to a and steady I is from b to a and increasing Physics 122B - Lecture 23 14 Energy in Inductors and Magnetic Fields A magnetic field stores considerable energy. Therefore, an inductor, which operates by creating a magnetic field, stores energy. Let’s consider how much energy UL is stored in an inductor L carrying current I: dI dI P I V I L LI dt dt I U L L IdI 12 LI 2 0 A solenoid of length l , area A, and N turns has L = 0N2A/l, so: U L LI 1 2 2 0 N 2 A 1 AlB 2 2 0 5/25/2017 2l 1 NI I Al 0 2 0 l uB 2 2 UL 1 2 B Al 20 U L LI 1 2 Physics 122B - Lecture 23 2 1 2 uB B 2 0 15 Energy Density in the Fields This is the magnetic analog of the energy stored in an electric field UE = e0 2 1 2 uB B 2 0 E 2 Magnetic Electric 5/25/2017 Physics 122B - Lecture 23 16 The LR Circuit VR VL 0 dI R dt I L RI L I dI 0 dt t dI R I I L 0 dt 0 I t ln L/ R I0 I (t ) I 0e t /( L / R ) 5/25/2017 Physics 122B - Lecture 23 17 Example: Exponential Decay in an LR Circuit The switch shown has been in position a for a long time. It changes to position b at t=0. a. What is the current in the circuit at t = 5 s? b. At what time has the current decayed to 1% of its original value? I 0 V / R (10 V) /(100 ) 100 mA I I 0e t / (100 mA)e (5 s) /(10 s) 61 mA L /( R R) (2.0 10-3 H) /(200 ) 10 s I (1 mA) t ln (10 s) ln 46 s (100 mA) I0 5/25/2017 Physics 122B - Lecture 23 18 Plumber’s RL Analogy Valve Constriction P1 P2 Pump The “plumber’s analogy” of an RL circuit is a pump (=battery) pumping water in a closed loop of pipe that includes a valve (=switch), a constriction (=resistor), and rotating flywheel turned by the flow. When the valve starts the flow, the flywheel begins to spin until a steady flow is achieved and the pressure difference across the flywheel (P2-P3) goes to zero. 5/25/2017 Physics 122B - Lecture 23 P3 Flywheel Pump = Battery Valve = Switch Constriction = Resistor Flywheel = Inductor Pressure = Potential Water Flow = Current 19 Capacitors: Early and Late* Initially, when a switch closes there is a potential difference of 0 across an uncharged capacitor. After a long time, the capacitor reaches its maximum charge and there is no current flow through the capacitor. Therefore, at t=0 the capacitor behaves like a short circuit (R=0), and at t=∞ the capacitor behaves like at open circuit (R=∞). Example: 100 V 100 V 12.5 A 100 V 2.5 A Circuit * From Lecture 16 5/25/2017 10 A 0 V at t=0 at t=∞ Calculate initial currents. IB = 100 V/8 = 12.5 A Calculate final potentials. Physics 122B - Lecture 23 20 Inductors: Early and Late Initially, when a switch closes an inductor appears to have an infinite resistance and has a maximum potential drop across it. Ultimately, the inductor reaches a steady current flow with no potential drop across it. Therefore, at t=0 the inductor behaves like at open circuit (R=∞), and at t=∞ the inductor behaves like a short circuit (R=0). This behavior is opposite that of a conductor. Example: Circuit at t=0 at t=∞ Calculate initial potentials. Calculate final currents. 5/25/2017 Physics 122B - Lecture 23 21 Question Rank the order of the time constants in these circuits. (a) 123 ; (d) 1>2>3 ; 5/25/2017 (b) 1<2<3 ; (e) 2>1>3 ; (c) 2<1<3 ; Physics 122B - Lecture 23 22 Lecture 23 Announcements Lecture HW has been posted and is due on Wednesday at 10 PM. Midterm examination 3 is this coming Friday--questions from lecture, tutorial AND Lab. 5/25/2017 Physics 122B - Lecture 23 23