Electric Field Lines. lecture 4.1.1

Transcription

1 Electric Field Lines Two protons, A and B, are in an electric field. Which proton has the larger acceleration? A. Proton A B. Proton B C. Both have the same acceleration. lecture 4.1.1

2 Electric Field Lines Two protons, A and B, are in an electric field. Which proton has the larger acceleration? A. Proton A B. Proton B C. Both have the same acceleration. lecture Stronger field where field lines are closer together. Weaker field where field lines are farther apart.

3 Chapter 6: Gauss s Law 6.1 Electric Flux 6.2 Explaining Gauss's Law 6.3 Applying Gauss's Law 6.4 Conductors in Electrostatic Carl Friedrich Gauss lecture 4.1.3

4 Symmetry of Charge Distributions Suppose we knew only 2 things about electric fields: The field points away from positive charges, toward negative charges. An electric field exerts a force on a charged particle. What could we deduce about the electric field of an infinitely long charged cylinder? lecture 4.1.4

5 Symmetry of Charge Distributions The field points away from positive charges, toward negative charges. An electric field exerts a force on a charged particle. lecture 4.1.5

6 Symmetry of Charge Distributions The field points away from positive charges, toward negative charges. An electric field exerts a force on a charged particle. lecture 4.1.6

7 Symmetry of Charge Distributions The field points away from positive charges, toward negative charges. An electric field exerts a force on a charged particle. lecture 4.1.7

8 Symmetry of Charge Distributions By symmetry arguments alone, an infinitely long charged cylinder must have a radial electric field. lecture 4.1.8

9 Symmetry of Charge Distributions lecture 4.1.9

10 Field Flux If you can t see into this box, but there is an outward-pointing electric field passing through every surface......it must contain a net positive charge. lecture What must be inside this box?

11 Field Flux What must be the net charge inside this box? a. b. c. d. positive negative zero cannot tell lecture

12 Field Flux Field flux (Φ) is how much field (E) passes through an area (A): The area vector: direction is perpendicular to surface magnitude equals the area lecture

13 Field Flux Field flux (Φ) is how much field (E) passes through an area (A): (see flux simulation on Bb) lecture

14 Field Flux The electric flux through the shaded surface is A. B. C. D N m/c 400 N m2/c Flux isn t defined for an open surface. lecture

15 Field Flux The electric flux through the shaded surface is A. B. C. D N m/c 400 N m2/c Flux isn t defined for an open surface. lecture

16 Field Flux The electric flux through the shaded surface is A. B. C. D N m/c 400 N m2/c Some other value. lecture

17 Field Flux The electric flux through the shaded surface is A. B. C. D N m/c 400 N m2/c Some other value. lecture

18 Field Flux The electric flux through the shaded surface is A. B. C. D. E cos20º N m2/c 400cos70º N m2/c 400 N m2/c Some other value. lecture

19 Field Flux The electric flux through the shaded surface is A. B. C. D. E cos20º N m2/c 400cos70º N m2/c 400 N m2/c Some other value. lecture

20 Field Flux To find the flux for a non-uniform field: Divide the surface into many small pieces of area δa. lecture

21 Field Flux To find the flux for a non-uniform field: Divide the surface into many small pieces of area δa. lecture

22 Field Flux To find the flux for a non-uniform field: Divide the surface into many small pieces of area δa. What is the flux through this surface? lecture Φe = 0

23 Field Flux For an electric field that is everywhere perpendicular to the surface and has the same magnitude E at every point: lecture

24 Field Flux lecture

25 Field Flux Surfaces A and B have the same shape and the same area. Which has the larger electric flux? A. B. C. D. Surface A has more flux. Surface B has more flux. The fluxes are equal. It s impossible to say without knowing more about the electric field. lecture

26 Field Flux Surfaces A and B have the same shape and the same area. Which has the larger electric flux? A. B. C. D. Surface A has more flux. Surface B has more flux. The fluxes are equal. It s impossible to say without knowing more about the electric field. lecture

27 Field Flux Which surface, A or B, has the larger electric flux? A. B. C. D. Surface A has more flux. Surface B has more flux. The fluxes are equal. It s impossible to say without knowing more about the electric field. lecture

28 Field Flux Which surface, A or B, has the larger electric flux? A. B. C. D. Surface A has more flux. Surface B has more flux. The fluxes are equal. It s impossible to say without knowing more about the electric field. lecture

29 Field Flux If the Gaussian surface is a closed surface: (we use the convention that the area vector da is defined to always point toward the outside) lecture

30 Field Flux These are cross sections of 3D closed surfaces. The top and bottom surfaces, which are flat, are in front of and behind the screen. The electric field is everywhere parallel to the screen. Which closed surface or surfaces have zero electric flux? A. B. C. D. E. Surface A Surface B Surface C Surfaces B and C All three surfaces lecture

31 Field Flux These are cross sections of 3D closed surfaces. The top and bottom surfaces, which are flat, are in front of and behind the screen. The electric field is everywhere parallel to the screen. Which closed surface or surfaces have zero electric flux? A. B. C. D. E. Surface A Surface B Surface C Surfaces B and C All three surfaces lecture

32 Field Flux For a single point charge: consider a Gaussian surface centered on the charge lecture

33 Field Flux For a single point charge: consider a Gaussian surface centered on the charge depends on the amount of charge, but not on the radius lecture

34 Field Flux The electric flux through any arbitrary closed surface surrounding a point charge q may be broken up into spherical and radial pieces. lecture

35 Field Flux The electric flux through any arbitrary closed surface surrounding a point charge q may be broken up into spherical and radial pieces. total flux same as through a single sphere: lecture

36 Field Flux The net electric flux is zero through a closed surface that does not contain any net charge. lecture

37 Field Flux Consider an arbitrary Gaussian surface and a group of charges q1, q2, q3, The contribution to the total flux for any charge qi inside the surface is qi /ϵ0. The contribution for any charge outside the surface is zero. lecture

38 Gauss s Law For any closed surface enclosing total charge Qin, the net electric flux through the surface is lecture

39 Gauss s Law The electric field is constant over each face of the box. The box contains A. B. C. D. Positive charge. Negative charge. No net charge. Not enough information to tell. lecture

40 Gauss s Law The electric field is constant over each face of the box. The box contains A. B. C. D. Positive charge. Negative charge. Net flux is inward. No net charge. Not enough information to tell. lecture

41 Gauss s Law Which spherical Gaussian surface has the larger electric flux? A. B. C. D. Surface A Surface B They have the same flux. Not enough information to tell. lecture

42 Gauss s Law Which spherical Gaussian surface has the larger electric flux? A. B. C. D. Total flux depends only on the enclosed charge, not the radius. Surface A Surface B They have the same flux. Not enough information to tell. lecture

43 Gauss s Law Spherical Gaussian surfaces of equal radius R surround two spheres of equal charge Q. Which Gaussian surface has the larger electric field? A. B. C. D. Surface A Surface B They have the same electric field. Not enough information to tell. lecture

44 Gauss s Law Spherical Gaussian surfaces of equal radius R surround two spheres of equal charge Q. Which Gaussian surface has the larger electric field? A. B. C. D. Surface A Surface B They have the same electric field. Not enough information to tell. lecture

45 Gauss s Law lecture

46 Gauss s Law 1. Gauss s law applies only to a closed surface, called a Gaussian surface. 2. A Gaussian surface is not a physical surface. It is an imaginary, mathematical surface that we define. 3. We can t find the electric field from Gauss s law alone. We need to apply Gauss s law in situations where, from symmetry and superposition, we already can guess the shape of the field. lecture

47 Gauss s Law A spherical Gaussian surface surrounds an electric dipole. The net enclosed charge is zero. Which is true? A. The electric field is zero everywhere on the Gaussian surface. B. The electric field is not zero everywhere on the Gaussian surface. C. Whether or not the field is zero on the surface depends on where the dipole is inside the sphere. lecture

48 Gauss s Law A spherical Gaussian surface surrounds an electric dipole. The net enclosed charge is zero. Which is true? A. The electric field is zero everywhere on the Gaussian surface. B. The electric field is not zero everywhere on the Gaussian surface. C. Whether or not the field is zero on the surface depends on where the dipole is inside the sphere. lecture The flux is zero, but that doesn t require the field to be zero.

49 Gauss s Law The electric flux is shown through two Gaussian surfaces. In terms of q, what are charges q1 and q2? A. q1 = 2q; q2 = q B. q1 = q; q2 = 2q C. q1 = 2q; q2 = q D. q1 = 2q; q2 = 2q E. q1 = q/2; q2 = q/2 lecture

50 Gauss s Law The electric flux is shown through two Gaussian surfaces. In terms of q, what are charges q1 and q2? A. q1 = 2q; q2 = q B. q1 = q; q2 = 2q C. q1 = 2q; q2 = q D. q1 = 2q; q2 = 2q E. q1 = q/2; q2 = q/2 lecture

51 Gauss s Law lecture

52 Gauss s Law For any closed surface enclosing total charge Qin, the net electric flux through the surface is lecture

53 Gauss s Law Use Gauss s Law to find the electric field near an infinite plane of charge with surface charge density σ (C/mC/m2). lecture

54 Gauss s Law A cylindrical Gaussian surface surrounds an infinite line of charge. The flux Φe through the two flat ends of the cylinder is A. B. C. D. E πrErE 2 πrer2e 2 rle It will require an integration to find out. lecture

55 Gauss s Law A cylindrical Gaussian surface surrounds an infinite line of charge. The flux Φe through the two flat ends of the cylinder is A. B. C. D. E πrErE 2 πrer2e 2 rle It will require an integration to find out. lecture

56 Gauss s Law A cylindrical Gaussian surface surrounds an infinite line of charge. The flux Φe through the wall of the cylinder is A. B. C. D. E. 0 2πrErLE πrer2le rle It will require an integration to find out. lecture

57 Gauss s Law A cylindrical Gaussian surface surrounds an infinite line of charge. The flux Φe through the wall of the cylinder is A. B. C. D. E. 0 2πrLErLE πrer2le rle It will require an integration to find out. lecture

58 Conductors & Electric Fields suppose you have a conductor (that may have charge on it) any electric field in the bulk of it would cause charges to move therefore: electric field is zero inside a conductor in equilibrium lecture

59 Conductors & Electric Fields suppose you have a conductor (that may have charge on it) any electric field in the bulk of it would cause charges to move therefore: electric field is zero inside a conductor in equilibrium consider a Gaussian surface just inside the surface of the conductor zero E means by Gauss s Law lecture therefore: all charge resides on the surface

60 Conductors & Electric Fields there is an electric field outside of a charged conducting surface but if it had any component parallel to the surface, charges would move therefore: electric fields just outside the surface of a conductor must be perpendicular to the surface lecture