8.022 (E&M) - Lecture 1

Transcription

1 8.0 (E&M) - Lecture 1 Topics: Gabriella Sciolla How is 8.0 organized? Brief math recap Introduction to Electrostatics Welcome to 8.0! 8.0: advanced electricity and magnetism for freshmen or electricity and magnetism for advanced freshmen? Advanced! Both integral and differential formulation of E&M Goal: look at Maxwell s equations and be able to tell what they really mean! Familiar with math and very interested in physics Fun class but pretty hard: 8.0 or 8.0T? 1

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3 Textbook E. M. Purcell Electricity and Magnetism Volume - Second edition Advantages: Bible for introductory E&M for generations of physicists Disadvantage: cgs units!!! 5 Problem sets Posted on the 8.0 web page on Thu night and due on Thu at 4:30 PM of the following week Leave them in the 8.0 lockbox at PEO Exceptions: Pset 0 (Math assessment) due on Monday Sep. 13 Pset 1 (Electrostatiscs) due on Friday Sep. 17 How to work on psets? Try to solve them by yourself first Discuss problems with friends and study group Write your own solution 6 3

4 Grades How do we grade 8.0? Homeworks and Recitations (5%) Two quizzes (0% each) Final (35%) Laboratory ( out of 3 needed to pass) NB: You may not pass the course without completing the laborator ies! More info on exams: Two in-class (6-100) quiz during normal class hours: Tuesday October 5 (Quiz #1) Tuesday November 9 (Quiz #) Final exam Tuesday, December 14 (9 AM - 1 Noon), location TBD All grades are available online through the 8.0 web page 7 Last but not least Come and talk to us if you have problems or questions 8.0 course material I attended class and sections and read the book but I still don t understand concept xyz and I am stuck on the pset! Math I can t understand how Taylor expansions work or why I should care about them Curriculum is 8.0 right for me or should I switch to TEAL? Physics in general! Questions about matter-antimatter asymmetry of the Universe, elementary constituents of matter (Sciolla) or gravitational waves (Kats) are welcome! 8 4

5 Your best friend in 8.0: math Math is an essential ingredient in 8.0 Basic knowledge of multivariable calculus is essential You must be enrolled in 18.0 or 18.0 (or even more advanced) To be proficient in 8.0, you don t need an A+ in 18.0 Basic concepts are used! Assumption: you are familiar with these concepts already but are a bit rusty Let s review some basic concepts right now! NB: excellent reference: D. Griffiths, Introduction to electrodynamics, Chapter

6 Derivative Given a function f(x), what is it s derivative? f df = dx x f The derivative tells us how fast f varies when x varies. x The derivative is the proportionality factor between a change in x and a change in f. What if f=f(x,y,z)? f f f df = dx + dy + dz x y z 11 Gradient Let s define the infinitesimal displacement f f f f f f df = dx + dy + dz =,,,, x y z x y z dl = dxx ˆ + dyy ˆ + dzz ˆ ( dx dy dz ) Definition of Gradient: f f f f f f grad f f ˆ x + ˆ + y ˆ z,, x y z x y z = f dl Conclusions: f measures how fast f(x,y,z) varies when x, y and z vary Logical extension of the concept of derivative! f is a scalar function but is a vector! f 1 6

7 The del operator Definition: x ˆ ˆ y ˆ z + +,, x y z x y z Properties: It looks like a vector It works like a vector But it s not a real vector because it s meaningless by itself. It s an operator. How it works: It can act on both scalar and vector functions: Acting on a sca l ar function: gradient f (vector) f Acting on a vector function with dot product: divergence (scalar) Acting on a sca l ar function with cross product: curl f (vector) 13 Divergence Given a vector function vxyz (,, ) vxyz (,, ) vx ˆ + vy ˆ + vz ˆ (, v, v ) v x y z x y z we define its divergence as: v v x y v z div v v + + x y z Observations: The divergence is a scalar Geometrical interpretat i on: it measures how much the funct i on spreads around a point. vxyz (,, ) 14 7

8 Divergence: interpretation Calculate the divergence for the following functions: v( x, y, z ) = xx ˆ + yy ˆ + ˆz v( x, y, z ) = zˆ v( x, y, z ) = x ˆ x y ˆ zˆ z div v=3>0 (faucet) div v=0 div v = -3 (sink) 15 Does this remind you of anything? Electric field around a charge has divergence.ne. 0! + - div E>0 for + charge: faucet div E <0 for charge: sink 16 8

9 Curl Given a vector function vxyz (,, ) vxyz (,, ) vx ˆ + vy y ˆ + vz ˆ (v, v, v ) x z x y z we define its curl as: xˆ ŷ ẑ v x y z Observations: The curl is a vector v v v x y z (,, Geometrical interpretation: it measures how much the function vxyz curls around a point. ) 17 Curl: interpretation Calculate the curl for the following function: vxy (,, z ) = yxˆ + xyˆ y x ˆ ˆ y ˆ z v = = x x y z y x 0 k ˆ This is a vortex: non zero curl! 18 9

10 Does this sound familiar? Magnetic fil ed around a wire : B I B 0 19 An now, our feature presentation: Electricity and Magnetism 10

11 The electromagnetic force: Ancient history 500 B.C. Ancient Greece Amber ( ελεχτρον= electron ) attracts light objects Iron rich rocks from µαγνεσια (Magnesia) attract iron C. F. du Fay: Two flavors of charges Positive and negative Priestley/Cavendish/Coulomb EM interactions follow an inverse square law: q Actual precision better than / q! F em r 1800 Volta Invention of the electric battery N.B.: Till now Electricity and Magnetism are disconnected! 1 The electromagnetic force: History (cont.) 180 Oersted and Ampere Established first connection between electricity and magnetism 1831 Faraday Discovery of magnetic induction 1873 Maxwell: Maxwell s equations The birth of modern Electro-Magnetism 1887 Hertz Established connection between EM and radiation 1905 Einstein Special relativity makes connection between Electricity and Magnetism as natural as it can be! 11

12 The electromagnetic force: Modern Physics! The Standard Model of Particle Physics Elementary constituents: 6 quarks and 6 leptons QUARKS u up d down c charm s strange t top b bottom LEPTONS νe electron neutrino e electron ν muon neutrino µ muon µ ντ tau neutrino τ tau Four elementary forces mediated by 5 bosons: Interaction Medi ator Relative Strength Range (cm) Strong Gluon El ectromagnetic Photon Infinite Weak Gravity W +/-, Z 0 Graviton? Infi nite 3 The electric charge The EM force acts on charges flavors: positive and negative Positive: obtained rubbing glass with silk Negative: obtained rubbing resin with fur Electric charge is quantized (Millikan) D1, D, D4 Multiples of the e = el ementary charge e = C (SI), esu (cgs) Q electron = -e; Q proton =+e Electric charge is conserved In any isolated system, the total charge cannot change If the total charge of a system changes, then it means the system is not isolated and charges came in or escaped. 4 1

13 Coulomb s law q q F = k ˆ r 1 r 1 1 Where: F ˆr 1 is the force that the charge q feels due to q is the unit vector going from q 1 to q Consequences: Newton s third law: F = F 1 1 Like signs repel, opposite signs attract 5 Units: cgs vs SI Units in cgs and SI (Sisteme Internationale) Length Mass Time Charge Current cgs cm g s electrostatic units (e.s.u.) e.s.u./s SI m Kg s Coulomb (C) Ampere (A) In cgs the esu is defined so that k=1 in Coulomb s law (1esu) 1 dyne = 1 esu = cm dyne (1cm) In SI, the Ampere is a fundamental constant k=1/(4πε 9 0 )= N C - m ε 0 =8.8x10-1 C N -1 m - is the permittivity of free space 6 13

14 Practical info: cgs - SI conversion table 3 =.9979 =c FAQ: why do we use cgs? Honest answer: because Purcell does 7 The superposition principle: discrete charges q 3 q q 1 q N Q q 4 q 5 The force on the charge Q due to all the other charges is equal to the vector sum of the forces created by the individual charges: qq q Q q Q q Q F r r r r i= N 1 N i ˆ ˆ ˆ ˆ Q = N = i r 1 r r N i = 1 r i 8 14

15 The superposition principle: continuous distribution of charges What happens when the distribution of charges is continuous? Take the limit for q dq and Σ integral: i q i r V Q q Q dq Q ρ dv Q ˆ ˆ r ˆ i=n i F Q = r i r = V V r i r i=1 r where ρ = charge per unit volume: volume charge density 9 The superposition principle: continuous distribution of charges (cont.) Charges are distributed inside a volume V: F Charges are distributed on a surface A: σ da Q F Q = ˆr A r Q = V ρ dv Q ˆr r Charges are distributed on a line L: λ dl Q F Q = ˆr L r Where: ρ = charge per un i t vo l ume: vo l ume charge dens i ty σ = charge per un i t area: surface charge dens i ty λ = charge per un it l ength: li ne charge dens i ty 30 15

16 Application: charged rod P: A rod of length L has a charge Q uniformly spread over i t. A test charge q is positioned at a distance a from the rod s midpoint. Q: What is the force F that the rod exerts on the charge q? a q L Qq Answer: F = y ˆ L a a + 31 Solution: charged rod Look at the symmetry of the problem and choose appropr ate coordinate system: rod on x axis, symmetric wrt x=0; a on y axis: i λ r L/ θ x dq=λdx q a L/ Symmetry of the problem: F // y axis; define =Q/L linear charge density Trigonometric relations: x/a=tg θ ; a=r cos θ dx=d θ /cos θ ; r=a/cos θ Consider the infinitesimal charge df y produced by the element dx: ad θ λ dx λq df y = df cos θ = q cos θ = λ q cos θ cosθ = cos θd θ r a a cos θ L / Now integrate between L/ and L/: λ q Qq F = ˆ y cos θ d θ = yˆ a L / L a a

17 Infinite rod? Taylor expansion! Q: What if the rod length is infinite? P: What does infinite mean? For al l practical purposes, infinite means >> than the other d i stances in the problem: L>>a: Qq Let s look at the solution: F = y ˆ L a a + l Tay or expand using (a/l) as expansion coefficient remembering that n nx n ( n 1) x (1 ± x ) = 1 ± + ±... for x <1 1!! and n nx n ( n + 1) x (1 ± x ) = for x <1 1!! λ Lq 1 a λ q a λ q 1 a λ q F = = 1 + = 1 + ~ 1 L a a L a... L a 1+ L 33 Rusty about Taylor expansions? Here are some useful reminders 34 17