Lecture 24 Chapter 31 Induction and Inductance Review Can produce an induced current and induced emf in a loop of wire when the number of magnetic field lines passing through the loop is changing Magnetic flux r r B B dA Faraday s law d B E dt Lenz s law An induced emf gives rise to a current whose B field opposes the change in flux that produced it Review Induced emf of a conductor moving with velocity v in a B field is given by E BLv Induced current in loop in a B field experiences a force r r FB iL B Found F1 opposes your r force Fapp r Fapp F1 Work you do in pulling the loop appears as thermal energy in the loop Inductance 20 Put a copper ring in a uniform B field which is increasing in time so the magnetic flux through the copper ring is changing By Faraday s law an induced emf and current are produced From Lenz s law their direction is counterclockwise If there is a current there must be an E field present to move the conduction electrons around ring Inductance 21 Induced E field is same as E field produced by static charges it will exert a force F qE on a charged particle Restate Faraday s law A changing B field produces an E field True even if no copper ring Inductance 22 Electric field lines produced by a changing B field are set of concentric circles If B field increasing or decreasing field lines are present decreasing have opposite direction If B field is constant with time no E field Inductance 23 Calculate work done on particle by induced E field Remember W q0E dW E dq For q0 moving along closed path the work is defined as W But FE is r r F ds FE q 0 E Equating equations for work gives r r q 0 E q0 E ds Inductance 24 Canceling q0 find that r r E E ds But from Faraday s law d B E dt So Faraday s law can be written as r r d B E ds dt A changing B field induces an E field Inductance 25 Inductor is a device used to produce a desired B field e g solenoid A current i in an inductor with N turns produces a magnetic flux B in its central region Inductance L is defined as L N B i SI unit is henry H 1H 1T m A 2 Inductance 26 N B L i What is inductance per unit length near the middle of a long solenoid First find flux of single loop in r r solenoid B B dA BA Remember turns N per unit length l is n N l nlBA L i Inductance 27 B field from a solenoid B 0in nlBA nl 0 in A 2 L l 0 n A i i Inductance per unit length is Depends only on geometry of device like capacitance L 2 0n A l Inductance 28 A changing current in a coil generates a self induced emf L in the coil Process is called self induction Change current in coil using a variable resistor EL will appear in coil only while the current is changing N B L i d B d N B d Li di EL N L dt dt dt dt Inductance 29 Induced emf only depends on rate of change of current not its magnitude Determine direction of EL using Lenz s law Self induced VL across inductor VL E L Ideal inductor Real inductor like real battery has some internal resistance V L E L iR di EL L dt Inductance 30 Checkpoint 4 Have an induced emf in a coil What can we tell about the current through the coil Is it moving right or left and is it constant decreasing or increasing Decreasing and rightward answer d OR Increasing and leftward answer e Inductance 31 RL circuit is a resistor and inductor in series Similar to RC circuit resistor capacitor in series Charging up a capacitor q C E 1 e t c Discharging capacitor q q0 e where t c C RC Inductance 32 Analogous time dependence on rise or fall of current if introduce an emf into or removie it from an RL circuit Initially close switch i is increasing through inductor so EL opposes rise i E i through R will be Long time later i is constant so EL 0 and i in circuit is R i E R Inductance 33 Initially an inductor acts to oppose changes in current through it Long time later inductor acts like ordinary conducting wire Apply loop rule right after switch has been closed at a Starting at x go clockwise di iR L E 0 dt Inductance 34 Differential equation similar to capacitors di E iR L dt Solution is E t L i 1 e R Where inductive time constant is L L R Satisfies conditions At t 0 i 0 At t i E E R Inductance 35 Now move switch to position b so battery is out of system Current will decrease with time and loop rule gives di iR L 0 dt Solution is E t L t L i e i0 e R Satisfies conditions E R At t 0 i i0 E At t i 0 L L R Inductance 36 Have a circuit with resistors and inductors What is the current through the battery just after close the switch Inductor oppose change in current through it Right after switch is closed current through inductor is 0 Inductor acts like broken wire Inductance 37 Apply loop rule E iR 0 Immediately after switch closed current through the battery is E i R Inductance 38 What is the current through the battery long time after the switch has been closed Currents in circuit have reached equilibrium so inductor acts like simple wire Circuit is 3 resistors in parallel E i R eq R eq R 3
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