Lecture 14.1 :! Electromagnetic Fields

Transcription

1 Lecture 14.1 :! Electromagnetic Fields Lecture Outline:! LR Circuits! E & B Transformations! The Displacement Current!! Textbook Reading:! Ch April 14,

2 Announcements Leo Anthony Soderberg arrived on April 9,

3 Announcements Homework #11 due on Tuesday, April 28, at 9am. Last homework assignment of the semester! Note that the due date is in 2 weeks!!!! Exam #3 on Thursday in class. Will cover Ch ! My office hours: today from 2-4pm, tomorrow (Wed.) 1-4pm.! Overview of the rest of the semester:! This week: reminder of latest Ch. 33 material, then onto Ch. 34.! Next week: Finish up Ch. 34 on Tuesday.! Lectures on April 23 and April 28 will be used to review course material.! Final Exam on Mon., May 4, 3-5pm in Watson Theatre (in Watson Hall). Corner of University Place and Comstock. You are allowed 4 sheets of notes, plus a calculator.! 3

4 Last Lecture... Maxwell proposed that a changing electric field induces a magnetic field (mirror idea to Faraday s law, which says changing magnetic field induces an electric field). 4

5 Last Lecture... Inductors are devices in circuits that can be used to store energy in magnetic fields (similar to Capacitors storing energy in electric fields). They have interesting behavior when placed in circuits. Inductance L I m 1 henry = 1 H 1Wb/A = 1 Tm 2 /A 5

6 Last Lecture... If the current through an inductor is changing, a potential difference develops across the inductor. Induced current Induced field V L = d m dt = L di dt We choose same sign convention as in resistors...voltage decreases in direction of current flow. 6

7 Last Lecture... We worked out the potential energy stored in the magnetic field of an inductor: U L = 1 A B 2 2µ 0 We could convert this to an energy density by dividing out the volume of the inductor. This result is actually very general. u B = 1 2µ 0 B 2 Recall for Electric Fields: u E = E 2 7

8 LC Circuits A circuit with an inductor and capacitor arranged in series is called an LC circuit. The current in this circuit will oscillate in time. 8

9 LR Circuits 9

10 Clicker Question #1 Sharon runs past Bill while holding a positive charge q. In Bill s reference frame, there is (or are)! A. Only an electric field. B. Only a magnetic field. C. An electric and a magnetic field. D. No fields. 10

11 Clicker Question #2 Sharon runs past Bill while holding a positive charge q. In Sharon s reference frame, there is (or are)! A. Only an electric field. B. Only a magnetic field. C. An electric and a magnetic field. D. No fields. 11

12 E & B Transformations Our current understanding of electric and magnetic fields can lead to some confusing scenarios: As Brittney runs by Alec, he sees a moving charge and hence a magnetic field, while Brittney doesn t see any magnetic field. If Alec creates a magnetic field that Brittney runs through, he expects a magnetic force on the charge, while she does not. 12

13 E & B Transformations Recall that inertial reference frames move at constant velocity with respect to one another. Newton s laws are valid in inertial frames. v CA = v CB + v BA dv CA dt = dv CB dt + dv BA dt a CA = a CB Observers in either frame agree on the acceleration of particle C, so they agree on the net force acting on the particle. 13

14 E & B Transformations Alec creates a region with zero electric field, and magnetic field B A. He observes a particle with velocity v CA traveling through this region to experience force F A. Brittney runs alongside the charge. What does she observe? 14

15 E & B Transformations Alec creates a region with zero electric field, and magnetic field B A. He observes a particle with velocity v CA traveling through this region to experience force F A. 15

16 E & B Transformations More generally, a charge can move through E and B fields with velocity v CA. We can imagine an observer moving at the same velocity as the charge. E B = E A + v CA B A Note: Frame B still can t say anything about possible magnetic fields since they are at rest with respect to the charge. 16

17 E & B Transformations What about transforming magnetic fields between frames? 17

18 Experimenters on earth have created the magnetic field shown. A rocket flies through the field, from right to left. Which are the field (or fields) in the rocket s reference frame? Clicker Question #3 18

19 Clicker Question #4 The E & B fields in frame A are shown. Which diagram shows the fields in frame B? 19

20 E & B Transformations A warning about relativity and Einstein... Fields at q 2 due to q 1 : E A = q 1 r 2 ĵ B A = µ 0 4 q 1 v CA r 2 ˆk 20

21 E & B Transformations We can now understand induced currents, no matter which frame of reference we choose. 21

22 The Displacement Current Recall Ampere s Law: B ds = µ 0 I through 22

23 The Displacement Current No rule saying we have to pick surface S 1. We could also choose to evaluate Ampere s Law through S 2, which is bounded by the same curve. 23

24 The Displacement Current Consider the circuit below with a capacitor and a battery. It appears we will get very different values for I through depending on which surface we use. 24

25 The Displacement Current We need to modify Ampere s Law to account for what s happening in surface S 2. This surface sees a changing electric flux. B d s = µ 0 I through + 0 d e dt 25

26 Gauss s Law for Magnetic Fields No Isolated Magnetic Poles (Monopoles)! POLE = POLE ( m ) closed surface = B da =0 26

27 Reminders! Exam #3 on Thursday. Tomorrow s recitation will be a time to practice/review for the exam.! After the exam, start reading Ch. 34.! HW #11 due Tuesday, April 28 at 9 am (in two weeks!) 27