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# Lect02 - Physics 212 Lecture 2 Today's Concept The Electric...

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Unformatted text preview: Physics 212 Lecture 2 Today's Concept: The Electric Field Continuous Charge Distributions Physics 212 Lecture 2, Slide 1 Physics Music BB Who is the Artist? A) B) C) D) E) Eric Clapton Bill Frisell Jimmy Page Jeff Beck Buddy Guy Why? Theme: guitar players (Englilsh) How about one of those guys listed last time? Nice acoustic album from 8 years ago… Physics 212 Lecture 2, Slide 2 Physics Your Comments What's the difference between the green and blue completion bars on the website? What exactly on is the Smartscore, and does it have any effect on my Class score? is all of it. it was like crawling the first day to tour de france today Some of the math magical calculations are a little confusing and some help on those Some some would be appreciated. That is really creepy - llogging all of our keystrokes and stuff! What else do you guys know? ogging That know? I would like to go over the infinite lines of charge as well as the equations and their meaning would for the physics presented. Especially lambda. Also, keeping rocking the good tunes! for 50 40 30 20 10 0 Confused 04 Confident How do X (distance of segment from origin) and L (length of line) relate? Isn't dx (length of segment) more related to L than dx is related to X? How is the integration of dE over L worked out, step by step? How do E and F relate conceptually? What is an electric field? How do charges exert this electric field?More questions and comments are to come. Physics 212 Lecture 2, Slide 3 Physics Coulomb’s Law (from last time) If there are more than two charges present, the total force If on any given charge is just the vector sum of the forces due vector of to each of the other charges: to q2 q2 F4,1 F4,1 F1 q1 F2,1 F1 F3,1 q3 F2,1 q3 F3,1 F4,1 F3,1 q4 F1 F2,1 q1 +q1 -> -q1 ﬂ direction reversed F2,1 q4 F1 F3,1 F4,1 MATH: MATH: F1 33 kq1q3 kq1q2 kq1q4 ˆ ˆ ˆ = 2 r12 + 2 r13 + 2 r14 r12 r13 r14 kq F1 kq2 kq ˆ ˆ ˆ = 2 r12 + 23 r13 + 24 r14 q1 r12 r13 r14 Physics 212 Lecture 2, Slide 4 Physics Electric Field “What exactly does the electric field that we calculate mean/represent? “ esent? “What is the essence of an electric field? “ F E≡ q The electric field E at a point in space is simply the force per unit charge at that point. Electric field due to a point charged particle Superposition Qi ˆ E = ∑ k 2 ri ri i Q ˆ E=k 2 r r q2 E4 E2 E Field points toward negative and Field Away from positive charges. E3 q4 q3 08 Physics 212 Lecture 2, Slide 5 Physics Preflights “I don't completely understand how to determine the direction of the electric field.” don't the 4) 70 60 B? 100 80 50 40 30 60 40 20 10 0 09 20 0 simulation Physics 212 Lecture 2, Slide 6 Physics Preflight E E 80 “The upper left +Q only affects the x direction in both and the lower right (+/-)Q only affects the y direction so in both, nothing cancels out, so they'll have the same magnitude. ” 60 40 20 0 12 1 2 Same Physics 212 Lecture 2, Slide 7 Physics BB Two Charges Two charges q1 and q2 are fixed at points (-a,0) and (a,0) as shown. Together they produce an electric field at point (0,d) which is directed along the negative y-axis. y (0,d) E (-a,0) q1 q2 (a,0) x Which of the following statements is true: a) b) c) d) 22 Both charges are negative Both charges are positive The charges are opposite There is not enough information to tell how the charges are There related related Physics 212 Lecture 2, Slide 8 Physics - + + + 23 - Physics 212 Lecture 2, Slide 9 Physics BB Preflight A B C INTERESTING: statement is correct, but given in support of “to the left” !! D “The force is proportional to the charge divided by The the square of the distance. Therefore, the force of the 2Q charge is 1/2 as much as the force of the Q charge. “ “Even though the charge on the right is larger, Even it is twice as far away, which makes the force it exherts on the test charge half that as the charge on the left, causing the charge to move to the right.” move 70 60 50 40 30 20 10 0 12 left right stay still The ratio between the R and Q on both sides is The 1:1 meaning they will result in the same magnitude of electric field acting in opposite directions, causing q to remain still. directions, Physics 212 Lecture 2, Slide 10 Physics BB Example “Show me more electric field examples, please!” +q What is the direction of the electric field at point P, the unoccupied corner of the square? P d -q +q d (A) Calculate E at point P. (D) know d (E) Need to know d & q Qi ˆ E = ∑ k 2 ri ri i 1 q Ex = − d2 4πεo 1 q Ey = − d2 4πεo 20 Need to (C) E = 0 (B) ( ( q 2d q 2d ) 2 ) 2 cos sin π 4 π 4 Physics 212 Lecture 2, Slide 11 Physics Continuous Charge Distributions “I don't understand the whole dq thing and lambda.” Summation becomes an integral (be careful with vector nature) Qi ˆ E = ∑ k 2 ri ri i dq ˆ E = ∫k 2 r r WHAT DOES THIS MEAN ?? Integrate over all charges (dq) r is vector from dq to the point at which E is defined Linear Example: λ = Q/L dE pt for E r charges 25 dq = λdx dq Physics 212 Lecture 2, Slide 12 Physics BB Charge Density “I would like to know more about the charge density.” • Linear (λ=Q/L) Coulombs/meter • Surface (σ = Q/A) Coulombs/meter2 • Volume (ρ = Q/ V) Coulombs/meter3 Some Geometry Asphere = 4πR 2 Acylinder = 2πRL Vsphere = 4 πR 3 Vcylinder = πR 2 L 3 What has more net charge?. A) A sphere w/ radius 2 meters and volume charge density ρ = 2 C/m3 B) A sphere w/ radius 2 meters and surface charge density σ = 2 C/m2 C) Both A) and B) have the same net charge. Q A = ρV = ρ 4 πR 3 3 QB = σA = σ 4πR 2 28 3 4 1ρ Q A ρ 3 πR = = R 2 QB σ 4πR 3σ Physics 212 Lecture 2, Slide 13 Physics Preflight 10) 70 60 50 40 “At point A the two fields will cancel each other out At but at point B the fields will act in the sme direction” but 30 20 10 0 29 Physics 212 Lecture 2, Slide 14 Physics BB Calculation “How is the integration of dE over L How y worked out, step by step?” worked Charge is uniformly distributed along the x-axis from the origin to x = a. The charge denisty is λ C/m. What is the x-componen t of the electric field at point P: (x,y) = (a,h)? P r h dq=λdx x x a We know: dq ˆ E = ∫k 2 r r What is (A) 33 dx x 2 dq r 2 ? dx (B) 2 a + h2 (C) λdx 2 a +h 2 (D) λdx 2 (a − x) + h 2 (E) λdx x2 Physics 212 Lecture 2, Slide 15 Physics Calculation dE dE x Charge is uniformly distributed along the x-axis from the origin to x = a. The charge denisty is λ C/m. What is the x-componen t of the electric field at point P: (x,y) = (a,h)? r θ1 We know: dq r2 = λdx (a − x) 2 + h 2 h θ2 x dq ˆ E = ∫k 2 r r θ2 P y BB x a dq=λdx E x = ∫ dE x What is dE x ? (A) dE cos θ1 33 (B) dE cos θ 2 (C) dE sin θ1 (D) dE sin θ 2 Physics 212 Lecture 2, Slide 16 Physics Calculation dE dE x Charge is uniformly distributed along the x-axis from the origin to x = a. The charge denisty is λ C/m. What is the x-componen t of the electric field at point P: (x,y) = (a,h)? r θ1 We know: dq r2 = λdx (a − x) 2 + h 2 h θ2 x dq ˆ E = ∫k 2 r r θ2 P y BB x a dq=λdx E x = ∫ dE x = ∫ dE cos θ 2 What is E x ? dx λ cos θ 2 ∞ (A) ∫ 4πεo − ∞ (a − x) 2 + h 2 λ cos θ 2 a dx (B) ∫ 4πεo 0 (a − x) 2 + h 2 (C) none of the above 33 cosθ2 DEPENDS ON x !! Physics 212 Lecture 2, Slide 17 Physics Calculation dE dE x Charge is uniformly distributed along the x-axis from the origin to x = a. The charge denisty is λ C/m. What is the x-componen t of the electric field at point P: (x,y) = (a,h)? r θ1 We know: dq r2 = h θ2 x dq ˆ E = ∫k 2 r r θ2 P y BB x a dq=λdx λdx E x = ∫ dE x = ∫ dE cos θ 2 (a − x) 2 + h 2 What is cos θ 2 ? (A) 33 x 2 a +h 2 (B) a−x 2 (a − x) + h 2 (C) a 2 a +h 2 (D) a (a − x) 2 + h 2 Physics 212 Lecture 2, Slide 18 Physics Calculation dE Charge is uniformly distributed along the x-axis from the origin to x = a. The charge denisty is λ C/m. What is the x-componen t of the electric field at point P: (x,y) = (a,h)? We know: dq r2 = dq ˆ E = ∫k 2 r r λdx dE x r θ1 h θ2 x x a dq=λdx E x = ∫ dE x = ∫ dE cos θ 2 (a − x) 2 + h 2 θ2 P y cos θ 2 = a−x (a − x) 2 + h 2 What is E x ( P) ? a−x λa E x ( P) = dx ∫ 4πεo 0 (a − x) 2 + h 2 ( 33 ) 3/ 2 h λ 1 − E x ( P) = 4πεo h h2 + a 2 Physics 212 Lecture 2, Slide 19 Physics Observation dE Charge is uniformly distributed along the x-axis from the origin to x = a. The charge denisty is λ C/m. What is the x-componen t of the electric field at point P: (x,y) = (a,h)? Note that our result can be rewritten more simply in terms of θ1. h λ 1 − E x ( P) = 4πεo h h2 + a 2 θ2 P y dE x r θ1 h θ2 x x a dq=λdx E x ( P) = λ (1 − sin θ1 ) 4πεo h Exercise for student: Change variables: write x in terms of θ Result: obtain simple integral in θ 33 λ π /2 E x ( P) = ∫ dθ cos θ 4πεo h θ 1 Physics 212 Lecture 2, Slide 20 Physics Notes • • • • • Preflight + Prelecture 3 due by 8:00 AM Tuesday August 31 Homework 1 is due Tuesday August 31 Labs start Monday August 30 Discussion Quiz next week will be on Coulomb’s Law and E Homework 2 is due Tuesday September 7 Physics 212 Lecture 2, Slide 21 Physics ...
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