Electricity & Magnetism Lecture 2: Electric Fields

Transcription

1 Electricity & Magnetism Lecture 2: Electric Fields Today s Concepts: A) The Electric Field B) Con3nuous Charge Distribu3ons Electricity & Magne3sm Lecture 2, Slide 1

2 Your Comments Suddenly, terrible haiku: Posi:ve test charge Repelled by op:mists Cup half electric a li@le bit concentra:on it requires. I had problems understanding the concept and calculation of charge on and infinite line??infinite lines of chargee??? it would help alot to go over those, everything else is simmplle Posi:ves and nega:ves, what do they really mean? It seems like we, as humans, just decided to name them this. define Electric field and electric force and charge in words please I almost forgot how exciting smartphysics was, yay Irony? Electricity & Magne3sm Lecture 2, Slide 2

3 Coulomb s Law (from last time) If there are more than two charges present, the total force on any given charge is just the vector sum of the forces due to each of the other charges: q 2 q 2 F 4,1 F 4,1 F 1 q 1 F 2,1 F 1 q 1 F 2,1 F 3,1 q 4 F 3,1 q 4 q 3 F 1 q 3 F 1 MATH: F 2,1 F 3,1 F 4,1 +q 1 > q 1 Þ direc:on reversed = F 2,1 F 3,1 F 4,1 Electricity & Magne3sm Lecture 2, Slide 3

4 In the diagrams below, the magnitude and direc:on of the electric field is represented by the length and direc:on of the blue arrows. Which of the diagrams best represents the electric field from a nega:ve charge?

5 3 charges are at the corners of an equilateral triangle Where is the E field 0? The black line is 1/2 way between the base and the top charge. A.Above the black line B. Below the black line C. on the black line D.nowhere except infinity E. somewhere else! Electric Field Line Applet Electric Field PhET

6

7 Electric Field What exactly does the electric field that we calculate mean/represent? What is the essence of an electric field? 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 par3cle Superposi3on E 4 q 2 Field points toward nega3ve and Away from posi3ve charges. E E 3 E 2 q 4 q 3 Electricity & Magne3sm Lecture 2, Slide 4

8 CheckPoint: Electric Fields1 x A +Q Q B Two equal, but opposite charges are placed on the x axis. The posi3ve charge is placed to the lev of the origin and the nega3ve charge is placed to the right, as shown in the figure above. What is the direc3on of the electric field at point A? o Up o Down o LeV o Right o Zero Electricity & Magne3sm Lecture 2, Slide 5

9 CheckPoint Results: Electric Fields1 x A +Q Q B Two equal, but opposite charges are placed on the x axis. The posi3ve charge is placed to the lev of the origin and the nega3ve charge is placed to the right, as shown in the figure above. What is the direc3on of the electric field at point A? o Up o Down o LeV o Right o Zero Electricity & Magne3sm Lecture 2, Slide 6

10 CheckPoint: Electric Fields2 x A +Q Q B Two equal, but opposite charges are placed on the x axis. The posi3ve charge is placed to the lev of the origin and the nega3ve charge is placed to the right, as shown in the figure above. What is the direc3on of the electric field at point B? o Up o Down o LeV o Right o Zero Electricity & Magne3sm Lecture 2, Slide 7

11 CheckPoint Results: Electric Fields2 x A +Q Q B Two equal, but opposite charges are placed on the x axis. The posi3ve charge is placed to the lev of the origin and the nega3ve charge is placed to the right, as shown in the figure above. What is the direc3on of the electric field at point B? o Up o Down o LeV o Right o Zero Polarization Demo Electricity & Magne3sm Lecture 2, Slide 8

12 CheckPoint: Magnitude of Field (2 Charges) In which of the two cases shown below is the magnitude of the electric field at the point labeled A the largest? +Q A +Q A Q +Q Case 1 Case 2 o Case 1 o Case 2 o Equal Electricity & Magne3sm Lecture 2, Slide 9

13 CheckPoint Results: Magnitude of Field (2 Chrg) In which of the two cases shown below is the magnitude of the electric field at the point labeled A the largest? +Q A +Q A E E Q +Q Case 1 Case 2 o Case 1 o Case 2 o Equal The upper lep +Q only affects the x direc:on in both and the lower right (+/- )Q only affects the y direc:on so in both, nothing cancels out, so they'll have the same magnitude. Electricity & Magne3sm Lecture 2, Slide 10

14 CheckPoint: Magnitude of Field (2 Charges) In which of the two cases shown below is the magnitude of the electric field at the point labeled B the largest? +Q B A +Q B A Q +Q Case 1 Case 2 o Case 1 o Case 2 o Equal Electricity & Magne3sm Lecture 2, Slide 9

15 Clicker Question: Two Charges Two charges q 1 and q 2 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 nega3ve y- axis. y (0,d) ( a,0) q 1 q 2 (a,0) x Which of the following statements is true: A) Both charges are nega3ve B) Both charges are posi3ve C) The charges are opposite D) There is not enough informa3on to tell how the charges are related Electricity & Magne3sm Lecture 2, Slide 11

16 + + + _ Electricity & Magne3sm Lecture 2, Slide 12

17 CheckPoint Results: Motion of Test Charge A posi3ve test charge q is released from rest at distance r away from a charge of +Q and a distance 2r away from a charge of +2Q. How will the test charge move immediately aver being released? o To the lev o To the right o Stay s3ll o Other The force is propor:onal to the charge divided by 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, it is twice as far away, which makes the force it exherts on the test charge half that as the charge on the lep, causing the charge to move to the right. The ra:o between the R and Q on both sides is 1:1 meaning they will result in the same magnitude of electric field ac:ng in opposite direc:ons, causing q to remain s:ll. Electricity & Magne3sm Lecture 2, Slide 13

18 Electric Field Example What is the direc3on of the electric field d at point P, the unoccupied corner of the square? q +q +q P d A) B) C) D) Need to know d E) Need to know d & q Calculate E at point P. 0 q E x = k B@ d 2 0 q E y = k B@ d 2 1 q p 2 cos 4 CA 2d 1 q p 2 sin 4 CA 2d Electricity & Magne3sm Lecture 2, Slide 14

19 About ˆr y ĵ î ~r 12 =r 12 cos î + r 12 sin ĵ x ~r 12 θ 2 ˆr 12 = r 12 cos î + r 12 sin ĵ r 12 ˆr 12 = cos î + sin ĵ 1 ˆr 12

20 For example ~r 12 =4î + 3 ĵ (1,2) ~r 12 ˆr 12 θ (5,5) r 12 = p = 5 cos = 4 5 sin = 3 5 ˆr 12 = r 12 cos î + r 12 sin ĵ r 12 ˆr 12 = 4 5 î ĵ

21 Continuous Charge Distributions I don't understand the whole dq thing and lambda. Summa3on becomes an integral (be careful with vector nature) Integrate over all charges (dq) WHAT DOES THIS MEAN? r is vector from dq to the point at which E is defined Linear Example: λ = Q/L pt for E r de charges dq = λ dx Electricity & Magne3sm Lecture 2, Slide 15

22 Clicker Question: Charge Density I would like to know more about the charge density. Linear (λ = Q/L) Coulombs/meter Surface (σ = Q/A) Coulombs/meter 2 Volume (ρ = Q/V) Coulombs/meter 3 Some Geometry What has more net charge?. A) A sphere w/ radius 2 meters and volume charge density ρ = 2 C/m 3 B) A sphere w/ radius 2 meters and surface charge density σ = 2 C/m 2 C) Both A) and B) have the same net charge. Electricity & Magne3sm Lecture 2, Slide 16

23 Prelecture Question A) What is the electric field at point a? B) C) D) E)

24 Clicker Question: Calculation How is the integra3on of de over L worked out, step by step? Charge is uniformly distributed along the x- axis from the origin to x = a. The charge density is λ C/m. What is the x- component of the electric field at point P: (x,y) = (a,h)? We know: y x dq = λ dx a r h P x What is? A) B) C) D) E) Electricity & Magne3sm Lecture 2, Slide 19

25 Clicker Question: Calculation Charge is uniformly distributed along the x- axis from the origin to x = a. The charge density is λ C/m. What is the x- component of the electric field at point P: (x,y) = (a,h)? y r P h θ 2 We know: θ 1 θ 2 x a dq = λ dx x What is? A) B) C) D) Electricity & Magne3sm Lecture 2, Slide 20

26 Clicker Question: Calculation Charge is uniformly distributed along the x- axis from the origin to x = a. The charge density is λ C/m. What is the x- component of the electric field at point P: (x,y) = (a,h)? y r P h θ 2 We know: θ 1 θ 2 x a dq = λ dx x What is? Z 1 dx k cos 2 k cos 1 (a x) 2 + h 2 2 A) B) Z a 0 dx (a x) 2 + h 2 C) none of the above cosθ 2 DEPENDS ON x! Electricity & Magne3sm Lecture 2, Slide 21

27 Clicker Question: Calculation Charge is uniformly distributed along the x- axis from the origin to x = a. The charge density is λ C/m. What is the x- component of the electric field at point P: (x,y) = (a,h)? y r P h θ 2 We know: θ 1 θ 2 x a dq = λ dx x What is? A) B) C) D) Electricity & Magne3sm Lecture 2, Slide 22

28 Calculation Charge is uniformly distributed along the x- axis from the origin to x = a. The charge density is λ C/m. What is the x- component of the electric field at point P: (x,y) = (a,h)? y r P h θ 2 We know: θ 1 θ 2 x a dq = λ dx x What is? E x (P) = k Z a 0 a x dx p (a x)2 + h 2 E x (P) = k 1! h p a2 + h 2 Electricity & Magne3sm Lecture 2, Slide 23

29 Observation Charge is uniformly distributed along the x- axis from the origin to x = a. The charge density is λ C/m. What is the x- component of the electric field at point P: (x,y) = (a,h)? y r P h θ 2 Note that our result can be rewri@en more simply in terms of θ 1. E x (P) = k! h 1 h p h2 + a 2 θ 1 θ 2 x a dq = λ dx E x (P) = k h (1 sin 1) x Exercise for student: Change variables: write x in terms of θ E x (P) = k h Z /2 d cos Result: obtain simple integral in θ 1 Electricity & Magne3sm Lecture 2, Slide 24