Physics 2102 Jonathan Dowling Physics 2102 Lecture 19 Ch 30 Inductors and RL Circuits Nikolai Tesla What are we going to learn A road map Electric charge Electric force on other electric charges Electric field and electric potential Moving electric charges current Electronic circuit components batteries resistors capacitors Electric currents Magnetic field Magnetic force on moving charges Time varying magnetic field Electric Field More circuit components inductors Electromagnetic waves light waves Geometrical Optics light rays Physical optics light waves Inductors Solenoids Inductors are with respect to the magnetic field what capacitors are with respect to the electric field They pack a lot of field in a small region Also the higher the current the higher the magnetic field they produce Capacitance how much potential for a given charge Q CV Inductance how much magnetic flux for a given current Li Using Faraday s law di EMF L dt Tesla m 2 Units L H Henry Ampere Joseph Henry 1799 1878 Self Inductance of a solenoid Solenoid of cross sectional area A length l total number of turns N turns per unit length n Field inside solenoid 0 n i Field outside 0 i B NAB NA 0ni Li 2 N L inductance 0 NAn 0 A l di EMF L dt Example The current in a 10 H inductor is decreasing at a steady rate of 5 A s If the current is as shown at some instant in time what is the magnitude and direction of the induced EMF a 50 V b 50 V i Magnitude 10 H 5 A s 50 V Current is decreasing Induced emf must be in a direction that OPPOSES this change So induced emf must be in same direction as current The RL circuit Set up a single loop series circuit with a battery a resistor a solenoid and a switch Describe what happens when the switch is closed Key processes to understand What happens JUST AFTER the switch is closed What happens a LONG TIME after switch has been closed What happens in between Key insights If a circuit is not broken one cannot change the CURRENT in an inductor instantaneously If you wait long enough the current in an RL circuit stops changing At t 0 a capacitor acts like a wire an inductor acts like a broken wire After a long time a capacitor acts like a broken wire and inductor acts like a wire RL circuits In an RL circuit while charging In an RC circuit while charging rising current emf Ldi dt and the Q CV and the loop rule mean loop rule mean charge increases from 0 to CE magnetic field increases from 0 to B current decreases from E R to 0 current increases from 0 to E R voltage across capacitor voltage across inductor increases from 0 to E decreases from E to 0 Example Immediately after the switch is closed what is the potential difference across the inductor a 0 V b 9 V c 0 9 V 10 9V 10 H Immediately after the switch current in circuit 0 So potential difference across the resistor 0 So the potential difference across the inductor E 9 V Example 3V Immediately after the switch is closed what is the current i through the 10 resistor a 0 375 A b 0 3 A c 0 40 10 Immediately after switch is closed current through inductor 0 Hence current trhough battery and through 10 resistor is i 3 V 10 0 3 A Long after the switch has been closed what is the current in the 40 resistor a 0 375 A Long after switch is closed potential b 0 3 A across inductor 0 c 0 075 A Hence current through 40 resistor 3 V 40 0 075 A 10 H Charging an inductor How does the current in the circuit change with time i di iR E L 0 dt Rt E L i 1 e R Time constant of RL circuit L R i t Small L R E R Large L R t Discharging an inductor The switch is in a for a long time until the inductor is charged Then the switch is closed to b i What is the current in the circuit Loop rule around the new circuit di iR L 0 dt E i e R Rt L i t Exponential discharge E R t Inductors Energy Recall that capacitors store energy in an electric field Inductors store energy in a magnetic field di E iR L dt di 2 iE i R Li dt Power delivered by battery i 2 d Li 2 iE i R dt 2 power dissipated by R d dt energy stored in L Example The switch has been in position a for a long time It is now moved to position b without breaking the circuit What is the total energy dissipated by the resistor until the circuit reaches equilibrium 10 9V 10 H When switch has been in position a for long time current through inductor 9V 10 0 9A Energy stored in inductor 0 5 10H 0 9A 2 4 05 J When inductor discharges through the resistor all this stored energy is dissipated as heat 4 05 J E 120V R1 10 R2 20 R3 30 L 3H What are i1 and i2 immediately after closing the switch What are i1 and i2 a long time after closing the switch What are i1 and i2 1 second after closing the switch What are i1 and i2 immediaately after reopening the switch What are i1 and i2 a long time after reopening the switch QuickTime and a Animation decompressor are needed to see this picture
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