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adapted from http://www.lab-initio.com/ Announcements Getting help at LEAD sessions: http://lead.mst.edu/ http://lead.mst.edu/request/tutorrequest.html http://lead.mst.edu/assist/index.html Announcements Other help: Disability Support Services (accommodation letters) http://dss.mst.edu/ Counseling Center http://counsel.mst.edu/ Testing Center http://testcenter.mst.edu/ Announcements Exam 1 is Tuesday, February 17, from 5:00-6:00 PM. According to the Student Academic Regulations “The period from 5:00 – 6:00 PM daily [is] to be designated for common exams. If a class or other required academic activity is scheduled during common exam time, the instructor of the class that conflicts with the common exam will provide accommodations for the students taking the common exam.” E-mail me by the end of Friday of next week if for some reason you have a non-class-related time conflict for Exam 1 (5:006:00 pm, Tuesday, February 17). Include in your email your recitation section letter and the details of your circumstance. (That doesn’t mean I accept responsibility for fixing the conflict.) Announcements Exam 1 is Tuesday, February 17, from 5:00-6:00 PM. The spring Career Fair is also February 17. The exam starts well after scheduled Career Fair activities have ended. If you have an interview or MUST attend a Career Fair activity during your recitation time on February 17, contact your recitation instructor. In such a case, you may wish to attend a recitation at a time other than your scheduled time. It is unlikely that Career Fair activities will conflict with the 5-6 PM exam time, but if there is such a conflict, e-mail me the details as soon as you find out about it. Announcements Homework for tomorrow: 1: 53, 90ab, 91a, 98 (also express the force in unit vector notation), 100 (reminder: all solutions must begin with starting equations) You only need to do part a of problem 91. You don’t need to be paranoid about using OSE’s, but don’t start a problem with a “random” equation from your book. Note the change. Announcements If a homework problem asks for the magnitude and direction of a vector, we are generally satisfied if you provide its components, or express it in unit vector notation. You might want to be paranoid and verify beforehand that this is OK. Announcements Labs begin this week. If you have lab this week, don’t miss it! 3L01 is “odd” 3L02 is “even” “Odd” labs meet this week (3L01, 3L03, 3L05, etc.). “Even” labs meet next week. You will need to purchase a lab manual. Go to the department office, room 102, to purchase your lab manual! The cost is $25.00. Do not ask me for a lab manual; I do not have them! http://campus.mst.edu/physics/courses/24lab/ Announcements For those who want to use Mastering Physics for practice, I “assigned” all of the problems from the chapters we will study this semester. Mastering Physics is OPTIONAL. You will get no course points for using it. You will not lose course points for not using it. Instructions for signing on are here: http://campus.mst.edu/physics/courses/24/CourseInformation/Mastering _Physics_for_Physics_2135.pdf. Once you open your Mastering Physics packet, you won’t be able to resell your text. Mastering Physics is based on the full text. Use the table here to see which full text chapters correspond to our text: http://campus.mst.edu/physics/courses/24/CourseInformation/physics_2 135_textbook.htm. Announcements There will be occasional errors in the lecture videos. After an error is discovered, it will be corrected or pointed out with a “callout” box in the video. It takes me about 15 hours to record and edit an hour’s worth of lecture video, so “polished” corrections will likely not be made until Summer 2015. The Big Picture, Part I In Lecture 1 you learned Coulomb's Law: 1 q1q 2 F = , 2 12 4πε 0 r12 r12 + - Q1 Q2 Coulomb’s Law quantifies the force between charged particles. The Big Picture, Part II In Lecture 1 you also learned about the electric field. There were two kinds of problems you had to solve… The Big Picture, Part IIa 1. Given an electric field, calculate the force on a charged particle. F E= q F= qE - F E You may not be given any information about where this electric field “comes from.” Or: given the force on a charged particle, determine the electric field that caused the force. The Big Picture, Part IIb 2. Given one or more charged particles, calculate the electric field they produce. We’ll focus on this topic today. q E=k 2 r q E=k 2 rˆ r I strongly recommend you start with this and do x and y components separately. Avoid for now. These are examples of “legal” equations: F = qE F E= q F= q E F E= q F = q E F F q = E E E E These are INCORRECT: q = = F F (do you know why?) q = E/F works *IF* q is positive. It is safer to write |q| = E/F. Today’s agenda: The electric field due to a charge distribution. You must be able to calculate electric field of a continuous distribution of charge. The Electric Field Due to a Continuous Charge Distribution (worked examples) finite line of charge general derivation: http://www.youtube.com/watch?v=WmZ3G2DWHlg ring of charge disc of charge infinite sheet of charge infinite line of charge semicircle of charge Instead of talking about electric fields of charge distributions, let’s work some examples. We’ll start with a “line” of charge. Example: A rod* of length L has a uniformly distributed total positive charge Q. Calculate the electric field at a point P located a distance d below the rod, along an axis through the left end of the rod and perpendicular to the rod. Example: A rod* of length L has a uniformly distributed total negative charge -Q. Calculate the electric field at a point P located a distance d below the rod, along an axis through the center of and perpendicular to the rod. I will work one of the above examples at the board in lecture. You should try the other for yourself. *Assume the rod has negligible thickness. Example: A rod of length L has a uniform charge per unit length and a total positive charge Q. Calculate the electric field at a point P along the axis of the rod a distance d from one end.* P d L To be worked at the board in lecture… *Assume the rod has negligible thickness. Example: calculate the electric field due to an infinite line of positive charge. There are two approaches to the mathematics of this problem. One approach is that of example 1.10. See notes here. An alternative mathematical approach is posted here. The result is 2k E 20 r r This is not an “official” starting equation! The above equation is not on your OSE sheet. In general, you may not use it as a starting equation! If a homework problem has an infinite line of charge, you would need to repeat the derivation, unless I give you permission to use it. Example: A ring of radius a has a uniform charge per unit length and a total positive charge Q. Calculate the electric field at a point P along the axis of the ring at a distance x0 from its center. To be worked at the blackboard in lecture. P x x0 Homework hint: you must provide this derivation in your solution to any problems about rings of charge (e.g. 1.53 or 1.55, if assigned). Visualization here (requires Shockwave, which downloads automatically): http://web.mit.edu/viz/EM/visualizations/electrostatics/calculatingElectricFields/RingIntegration/RingIntegration.htm y Ex a P x0 x x kx 0 Q 2 0 a 2 3/2 Ey Ez 0 E x0 is positive! Also “legal” answers: E kx 0 Q 2 2 x a 0 ˆi 3/2 E kx 0 Q 2 x 0 a 2 3/2 , away from the center These equations are only valid for P along the positive x-axis! Awesome Youtube Derivation: http://www.youtube.com/watch?v=80mM3kSTZcE (he leaves out a factor of a in several steps, but finds it in the end). What would be different if Q were negative? If P were on the negative x-axis? Example: A disc of radius R has a uniform charge per unit area . Calculate the electric field at a point P along the central axis of the disc at a distance x0 from its center. P R x0 x Example: A disc of radius R has a uniform charge per unit area . Calculate the electric field at a point P along the central axis of the disc at a distance x0 from its center. The disc is made of concentric rings. r P R x x0 2r dr The ring has infinitesimal thickness, so you can imagine it as a rectangular strip. Imagine taking a ring and cutting it so you can lay it out along a line. The length is 2r, the thickness is dr, so the area of a ring at a radius r is 2rdr. Caution! In the previous example, the radius of the ring was R. Here the radius of the disc is R, and the rings it is made of have (variable) radius r. Example: A disc of radius R has a uniform charge per unit area . Calculate the electric field at a point P along the central axis of the disc at a distance x0 from its center. dq The charge on each ring is dq = (2rdr). Let’s assume is positive so dq is positive. r charge on ring = charge per area × area P R x0 x dEring We previously derived an equation for the electric field of this ring. We’ll call it dEring here, because the ring is an infinitesimal part of the entire disc. dE ring kx 0 dq ring x 2 0 r 2 3/2 Example: A disc of radius R has a uniform charge per unit area . Calculate the electric field at a point P along the central axis of the disc at a distance x0 from its center. Let’s assume is positive so dq is positive. dq dE ring r P x0 R E disc dE disc ring disc kx 0 2rdr x x 2 0 r 2 3/2 x dEring kx 0 dq ring 2 0 r 2 3/2 kx 0 R 0 x kx 0 ( 2rdr ) x 2 0 r 2r dr 2 0 r 2 3/2 2 3/2 Let’s assume is positive so dq is positive. dq r P R x0 x dEring E disc kx 0 R 0 x 2r dr 2 0 r 2 3/2 x2 r 0 kx 0 1/ 2 2 1/2 E disc You know how to integrate this. The integrand is just (stuff)-3/2 d(stuff) R x0 2k x 0 x 0 x 2 R 2 1/2 0 0 Kind of nasty looking, isn’t it. P R x0 x Edisc x0 x0 E x 2k x 0 x 2 R 2 1/2 0 As usual, there are several ways to write the answer. E y Ez 0 x0 x0 ˆi E 2k x 0 x 2 R 2 1/2 0 Or you could give the magnitude and direction. Example: Calculate the electric field at a distance x0 from an infinite plane sheet with a uniform charge density . An infinite sheet is “the same as” disc of infinite radius. Esheet x x 0 lim 2k 0 R x 0 x 2 R 2 1/2 0 1 Take the limit and use k to get 4 0 Esheet 20 . This is the magnitude of E. The direction is away from a positively-charged sheet, or towards a negatively-charged sheet. Example: Calculate the electric field at a distance x0 from an infinite plane sheet with a uniform charge density . Esheet 20 . Interesting...does not depend on distance from the sheet. Does that make sense? This is your fourth Official Starting Equation, and the only one from all of today’s lecture! I’ve been Really Nice and put this on your starting equation sheet. You don’t have to derive it for your homework! Example: calculate the electric field at “center” of semicircular line of uniformly-distributed positive charge, oriented as shown. +Q R y You don’t have to follow the steps in the exact order I present here. Just let the problem tell you what to. You may do things in a different order; that’s probably OK. x ds To be worked at the blackboard in lecture. d R dE Help! We covered a lot of material in a brief time. If you want to explore a slightly different presentation of this at your leisure, try the MIT Open Courseware site. http://ocw.mit.edu/courses/physics/8-02sc-physics-ii-electricity-andmagnetism-fall-2010/index.htm E&M main page http://www.youtube.com/watch?v=OsWDUqJQcpk&feature=relmfu lecture on electric fields http://ocw.mit.edu/courses/physics/8-02sc-physics-ii-electricity-andmagnetism-fall-2010/electric-fields/ all course material on electric fields Homework Hints (may not apply every semester) Your starting equations so far are: q1q 2 F k 2 12 r12 F0 E= q0 q E=k 2 r Esheet 20 . (plus Physics 23 starting equations). This is a “legal variation” (use it for charge distributions): dq dE=k 2 . r You can remove the absolute value signs if you know that dq is always positive. Homework Hints (may not apply every semester) Suppose you have to evaluate this integral in your homework… a dx a r x 0 2 Let u=(a+r-x)2. Then du=-dx and u=a+r when x=0, u=r when x=a. Or look it up if you have tables that contain it. Homework Hints (may not apply every semester) The integrals below are on page 522 of your text. dx x 2 a 2 3 2 x dx x 2 a 2 3 2 Your recitation instructor will supply you with needed integrals. The above integrals may or may not be needed this semester. Learning Center Today 2:00-4:30, 6:00-8:30 Rooms 129/130 Physics