Gabriella Sciolla 8.022 (E&M) -Lecture 1 Topics: How is 8.022 organized? Brief math recap Introduction to Electrostatics 2 Welcome to 8.022! Advanced! September 8, 2004 8.022 – Lecture 1 8.022: advanced electricity and magnetism for freshmen or electricity and magnetism for advanced freshmen? 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: 8.022 or 8.02T? 1 Textbook E. M. PurcellElectricity and Magnetism Volume 2 -Second edition Advantages: Bible for introductory E&M for generations of physicists Disadvantage: cgs units!!! September 8, 2004 8.022 – Lecture 1 5 Problem sets Posted on the 8.022 web page on Thu night and due on Thu at 4:30 PM of the following week Leave them in the 8.022 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 September 8, 2004 8.022 – Lecture 1 6 3 7 Grades How do we grade 8.022? NB: You may not ies! More info on exams: Tuesday October 5 (Quiz #1) Final exam September 8, 2004 8.022 – Lecture 1 Homeworks and Recitations (25%) Two quizzes (20% each) Final (35%) Laboratory (2 out of 3 needed to pass) pass the course without completing the laborator Two in-class (26-100) quiz during normal class hours: Tuesday November 9 (Quiz #2) Tuesday, December 14 (9 AM -12 Noon), location TBD All grades are available online through the 8.022 web page 8 …Last but not least… Math care about them… Curriculum September 8, 2004 8.022 – Lecture 1 Come and talk to us if you have problems or questions 8.022 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! I can’t understand how Taylor expansions work or why I should is 8.022 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! 4 10 Your best friend in 8.022: math To be proficient in 8.022, you don’t need an A+ in 18.022 September 8, 2004 8.022 – Lecture 1 Math is an essential ingredient in 8.022 Basic knowledge of multivariable calculus is essential You must be enrolled in 18.02 or 18.022 (or even more advanced) Basic concepts are used! Assumption: you are familiar with these concepts already but are a bit rusty… NB: excellent reference: D. Griffiths, Introduction to electrodynamics, Chapter 1. Let’s review some basic concepts right now! 5 11 Derivative tells us how fast f fdf dx x ∂ = ∂ f x ∂ ∂ f f fdf dx dy dz x y z ∂ ∂ ∂ = + + ∂ ∂ ∂ What if September 8, 2004 8.022 – Lecture 1 Given a function f(x), what is it’s derivative? The derivative varies when x varies. The derivative is the proportionality factor between a change in x and a change in f. f=f(x,y,z)? 12 Gradient ( ), , , , f f f f f fdf dx dy dz x y z x y z ⎛ ⎞∂ ∂ ∂ ∂ ∂ ∂ = + + = •⎜ ⎟∂ ∂ ∂ ∂ ∂ ∂⎝ ⎠ ˆ ˆ ˆdl dxx dzz= + + f is a vector! =∇• ˆ ˆ ˆ , , f f f f f ff f x y z x y z x y z ⎛ ⎞∂ ∂ ∂ ∂∂∂+ + ≡⎜ ⎟∂ ∂ ∂ ∂ ∂ ∂⎝ ⎠ f∇ f∇ September 8, 2004 8.022 – Lecture 1 Let’s define the infinitesimal displacement dx dy dz dyy Definition of Gradient: Conclusions: measures how fast f(x,y,z) varies when x, y and z vary Logical extension of the concept of derivative! is a scalar function but fdl grad ≡∇ ≡ 6 13 The “del” operator Properties: How it works: l (vector) (scalar) l ˆ ˆ ˆ , ,x y z x y z ⎛ ⎞ ⎛ ⎞∂ ∂ ∂ ∂ ∂ ∂∇≡ + + ≡⎜ ⎟ ⎜ ⎟∂ ∂ ∂ ∂ ∂ ∂⎝ ⎠ ⎝ ⎠ f∇ f∇• f∇× x y z September 8, 2004 8.022 – Lecture 1 Definition: 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. It can act on both scalar and vector functions: Acting on a sca ar function: gradient Acting on a vector function with dot product: divergence Acting on a sca ar function with cross product: curl (vector) 14 Divergence we define its divergence as: Observations: The divergence is a scalar i i “spreads around a point”. div yx z vv v v v x y z ∂∂ ∂≡ + + ∂ ∂ ∂ ˆ ˆ ˆ) ( )x y z x y z≡ + + ≡ ) ) September 8, 2004 8.022 – Lecture 1 Given a vector function Geometrical interpretat on: it measures how much the funct on ≡∇• (, , , , vxyz vx vy vz v v v (, , vxyz (, , vxyz 7 15 Divergence: interpretation ˆ ˆ ˆ) z= + + div v=3>0 (faucet) ˆ) z= div v=0 ˆ ˆ ˆ) x z=−− −xx y zSeptember 8, 2004 8.022 – Lecture 1 Calculate the divergence for the following functions: (,, vxyz yy z (, , vx yz (,, vxyz x y div v = -3 (sink) 16 Does this remind you of anything? + -September 8, 2004 8.022 – Lecture 1 Electric field around a charge has divergence .ne. 0 ! div E <0 for – charge: sink div E>0 for + charge: faucet 8 Curl (, , Given a vector function vxyz ) (, , ˆ+ y , , vxyz )≡vx vy ˆ+vz ˆ≡(v v v )x z xyz we define its curl as: xˆ yˆ zˆ ∂∂∂∇×v≡ ∂x ∂y ∂z v vvx y zObservations: The curl is a vector (, , Geometrical interpretation: it measures how much the function vxyz ) “curls around a point”. September 8, 2004 8.022 – Lecture 1 17 18 Curl: interpretation ˆ ˆ)vxyz yxxy=−+ ˆ ˆ ˆ ˆ2 0 x y z v k x y z y x ∂ ∂ ∂∇× = = ∂ ∂ ∂ − x y September 8, 2004 8.022 – Lecture 1 Calculate the curl for the following function: This is a vortex: non zero curl! (, , 9 19 Does this sound familiar? Magnetic fil I B 0B∇× ≠ September 8, 2004 8.022 – Lecture 1 ed around a wire : An now, our feature presentation: Electricity and Magnetism 10 21 The electromagnetic force: Ancient history… Amber ( Actual precision better than 2/109! 1 2 2em qqF r ∝ September 8, 2004 8.022 – Lecture 1 500 B.C. – Ancient Greece ελεχτρον=“electron”) attracts light objects Iron rich rocks from µαγνεσια (Magnesia) attract iron 1730 -C. F. du Fay: Two flavors of charges Positive and negative 1766-1786 – Priestley/Cavendish/Coulomb EM interactions follow an inverse square law: 1800 – Volta Invention of the electric battery N.B.: Till now Electricity and Magnetism are disconnected! The electromagnetic force: …History… (cont.) 1820 – 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! September 8, 2004 8.022 – Lecture 1 22 11 23 The electromagnetic force: Modern Physics! 10-131037Gluon Infi1035El 1 1024 Relative Strength Infi 10-15 Medi Graviton?Gravity W+/-, Z0Weak September 8, 2004 8.022 – Lecture 1 The Standard Model of Particle Physics Elementary constituents: 6 quarks and 6 leptons Four elementary forces mediated by 5 bosons: Strong nite Photon ectromagnetic nite Range (cm) ator Interaction 24 The electric charge Multiples of the e = el e Q = -e; Q =+e Electric charge is conserved September 8, 2004 8.022 – Lecture 1 The EM force acts on charges 2 flavors: positive and negative Positive: obtained rubbing glass with silk Negative: obtained rubbing resin with fur Electric charge is quantized (Millikan) ementary charge = 1.602 10-19 C (SI), 4.803 10-10 esu (cgs) electron proton 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. D1, D2, D4 12 uup c charm t top d QUARKS down s strange b bottom n electron neutrino LEPTONS e n muon neutrino m ntau neutrino t electron muon m tau t e25 Coulomb’s law Where: 2 1 1 to q2 Consequences: 1 2 2 2 21 ˆ | | qqF k r r = 21ˆr 2F 2 1F F= − September 8, 2004 8.022 – Lecture 1 is the force that the charge q feels due to q is the unit vector going from q Newton’s third law: Like signs repel, opposite signs attract 2 1 26 Units: cgs vs SI k=1/(4πε0 9 N C-2 m2 ε0=8.8x10-12 C2 N-1 m-2 Coulomb (C)eCharge mcmLength Kgg Ampere (A) s SI Current s 2 2(1cm) → September 8, 2004 8.022 – Lecture 1 Units in cgs and SI (Sisteme Internationale) In cgs the esu is defined so that k=1 in Coulomb’s law In SI, the Ampere is a fundamental constant )=8.99 10 is the permittivity of free space lectrostatic units (e.s.u.) Mass cgs e.s.u./s Time (1esu) 1 dyne = 1 esu = cm dyne 13 27 September 8, 2004 8.022 – Lecture 1 Practical info: cgs -SI conversion table FAQ: why do we use cgs? Honest answer: because Purcell does… “3”=2.9979… =c 28 Q q5 q4 q3 q2 q1 qN 1 2 1 22 2 2 2 11 2 ˆ ˆ ˆ ˆ... | | | | | | | | iN N i Q N i iN i qQqQF r r r r r r r r = = = + + + = ∑ September 8, 2004 8.022 – Lecture 1 The superposition principle: discrete charges qQ qQ 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: 14 29 continuous distribution of charges where ρ i i2 2 2V V i q Q dq Q ρˆ ˆ ˆr r r |r | |r| |r|QF = → =∑ ∫ ∫ idq and Σ integral: Q qi r V September 8, 2004 8.022 – Lecture 1 The superposition principle: = charge per unit volume: “volume charge density” i=N i=1 dV Q What happens when the distribution of charges is continuous? Take the limit for q 30 Charges are distributed inside a volume V: Charges are distributed on a surface A: Charges are distributed on a line L: continuous distribution of charges (cont.) Where: ρ i l l i σ i i λ it l li i 2A ˆr |r|QF σ = ∫ 2V ρ ˆr |r|QF = ∫ 2L ˆr |r|QF λ = ∫ September 8, 2004 8.022 – Lecture 1 The superposition principle: = charge per un t vo ume: “vo ume charge dens ty” = charge per un t area: “surface charge dens ty” = charge per un ength: “ ne charge dens ty” da Q dV Q dl Q 15 31 Application: charged rod i L q a Answer: 2 2 ˆ 2 QqF y L aa = ⎛⎞+⎜⎟⎝⎠ September 8, 2004 8.022 – Lecture 1 P: A rod of length L has a charge Q uniformly spread over 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? 32 Solution: charged rod i λ θ θ θ θ θ y L/2 q L/2 θ dq=λdx a x r 2 22 2 cosy ad dx qdF dF q q d ar a θ λ λθθ θλ θ θθ θ = = = = /2 2 /2 2 ˆ ˆ 2 L L qF y d ya L aa λ θθ − = = ⎛⎞+⎜⎟⎝⎠ ∫ September 8, 2004 8.022 – Lecture 1 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: Symmetry of the problem: F //y axis; define =Q/L linear charge density Trigonometric relations: x/a=tg ; a=r cos dx=d /cos2 ; r=a/cos Consider the infinitesimal charge dF produced by the element dx: Now integrate between –L/2 and L/2: cos cos cos cos cos cos Qq 16 33 Infinite rod? Taylor expansion! l i l 2 2 2 ˆ 2 QqF y L aa = ⎛⎞+⎜⎟⎝⎠ 2 2 2 2 ( 1)(1 ) 1 1! and ( 1)(1 ) 1 1! n n nx x x nx x x − − ± +± = +∓ ∓ 1 2 22 1 2 2 1 21 1 ~ 2 2 2 2212 Lq q a q a qaF a L a L aL a L λ λ λ λ −⎛ ⎞ ⎛ ⎞ = = + = − +⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠⎛ ⎞+⎜ ⎟⎝ ⎠ September 8, 2004 8.022 – Lecture 1 Q: What if the rod length is infinite? P: What does “infinite” mean? For al practical purposes, infinite means >> than the other d stances in the problem: L>>a: Let’s look at the solution: Tay or expand using (2a/L) as expansion coefficient remembering that ... for x <1 2! ... for x <1 2! n n n n ± =±+ ... ⎛⎞ ⎛⎞ ⎜⎟ ⎜⎟ ⎝⎠ ⎝⎠ 34 Here are some useful reminders… September 8, 2004 8.022 – Lecture 1 Rusty about Taylor expansions? 17