Chapter 16 Electric Field

1. Chapter 16 Electric Field

2. Main Points of Chapter 16 • Electric field • Superposition • Electric dipole • Electric field lines • Field of a continuous distribution of charge • Motion of a charge in a field • Electric dipole in external electric field

3. 16-1 Electric Field 1. Electric Field Early：an “action-at-a-distance” force Later:Faraday introduced “Electric Field ” E field charge Charge Electric field

4. 2. Definition of Electric field Field Point Size is small enough Test charge Charge is very small source charge Test charge must be small enough that it does not affect field:

5. SI N/C (or V/m) The electric field is defined: Some values of Electric fields: atmosphere at earth’s surface in clear weather 100-200 N/C value sufficient to cause electrical breakdown in dry air

6. For a point charge Therefore, the electric field of a point charge is: The field points outwards from a positive charge and inwards to a negative one:

7. Comparison of Electric Force with Electric Field • Electric Force (F) - the actual force felt by a charge at some location. • Electric Field (E) - found for a location only – tells what the electric force would be if a charge were located there: A useful concept

8. the unit vector directed from the position of the kth charge to the point where the field is evaluated 3. Principle of Superposition of Electric Fields Since electric forces add by superposition, the electric field does as well. For a group of point charges:

9. y x • Act What is the electric field at the origin due to this set of charges? a a Solution +q +q a a Notice that the fields from the top-right and bottom left cancel at the origin a +q

10. 4. Electric Dipoles and Their Fields An electric dipole is defined as equal and opposite charges a distance l apart: electric dipole moment is defined as and points from the negative charge towards the positive one.

11. Electric dipoles are often found in nature. Charged objects can induce electric dipoles (left); molecules may have permanent electric dipole moments due to their structure (right).

12. Example: Find the electric field of the electric dipole at point a and point b. b Solution r a O x

13. At any point far from a dipole in any direction Similarly, for point b, b r

14. ACT The electric field of a dipole at distance large compared to the charge separation ________ • A) decreases linearly with increasing distance. • B) remains constant as the distance increases. • C) decreases inversely with the cube of the distance. • D) decreases inversely with the square of the distance. • E) cannot be determined.

15. 16-2 Electric Field Lines Electric field lines are a useful aid to visualizing the electric field. There are two rules to drawing these lines: • The electric field is tangent to the field line at every point. • 2. The density of electric field lines is an indicator of relative field strength. The next slide shows field lines for a point charge, including the decrease in density as one moves farther from the charge.

16. Field lines of a positive point charge:

17. Left: two equal, same-sign charges Right: an electric dipole Important note: We always draw only a few sample field lines; otherwise the sketch would be solid color.

18. Field Lines oftwo equal, same-sign charges • There is a zero halfway between the two charges • r >> a: looks like the field of point charge (+2q)

19. Electric field lines have certain properties which should be carefully noted: • Lines leave (+) charges and return to (-) charges, • never discontinue in empty space • Tangent of line = direction of E • Local density of field lines µ local magnitude of E • No two field lines ever cross, even when multiple charges are present.

20. 1) 2) 3) 4) 5) no way to tell • ACT What are the signs of the charges whose electric fields are shown at right? Electric field lines originate on positive charges and terminate on negative charges. Which of the charges has the greater magnitude? The red one

21. 16-3 The Field of a Continuous Distribution To find the field of a continuous distribution of charge, treat it as a collection of near-point charges: Summing over the infinitesimal fields:

22. Finally, making the charges infinitesimally small and integrating rather than summing:  : linear charge density  : surface charge density  : volume charge density

23. dy r • Example An infinitely long wire is uniformly charged. The charge density is l .Find the Electric field at point P on the x-axis at x=x0 Solution y P x0 x

24. - decreases as + as seen from above The Electric Field produced by an infinite line of charge is: - everywhere perpendicular to the line - is proportional to the charge density next lecture: Gauss’ Law makes this trivial!!

25. ACT A long line of charge with charge per unit length λ1 is located on the x-axis and another long line of charge with charge per unit length λ2 is located on the y-axis with their centers crossing at the origin. In what direction is the electric field at point z = a on the positive z-axis if λ1 and λ2 are positive? • A) the positive z-direction • B) halfway between the x-direction and the y-direction • C) the negative z-direction • D) all directions are possible parallel to the x-y plane • E) cannot be determined

26. Act The figure here shows three nonconducting rods, one circular and two straight. Each has a uniform charge of magnitude Q along its top half and another along its bottom half. For each rod, what is the direction of the net electric field at point P?

27. dq  • Example A uniformly charged circular arc has radius R and subtends an angle 2q0.The total charge is q. Calculate the electric field at point P . y Solution P x R

28. x r  dq O • Act: Find the electric field at point P on the axis of a uniformly charged ring of total charge q. The radius of the ring is R, the distance from P to the center of the ring is x. Solution P R • whenx = 0（P is at the center of the ring) • whenx>>R

29. x O • Act: Find the electric field at a distance xalong the axis of a uniformly charged circular disk of radius R and charge Q Solution P r R

30. x O P If x>>R r R Field of a point charge If R>>x Field of an infinite uniformly charged plane

31. y dy q1 x x as seen from above Field of an infinite uniformly charged plane Divide the plane into narrow straight strips s q1 x x

32. The result: The individual fields: The superposition: From the electric field due to a uniform sheet of charge, we can calculate what would happen if we put two oppositely-charged sheets next to each other:

33. Point Charge ~ 1/r2 Dipole ~ 1/r3 Infinite Line of Charge ~ 1/r infinite uniformly charged plane ~ 1/r0 Summary of Electric Field Lines

34. Act A large flat has a uniform charge density s . A small • circular hole of radius R has been cut in the middle of the • surface. Calculate the electric field at point P(x) Solution Think that the configuration is composed of one uniformly charged plane with charge density +s and a uniformly charged circular disk -s R x . P Use superposition x

35. r dz z • Example: Find the electric field at point P on the central axis of the solid cone. The total charge is q Solution z P H R y O x

36. q 16-4 Motion of a Charge in a Field Deflection of Moving Charged Particles( electrons) We can control the motion of a beam of charged particles

37. ACT An electron beam moving horizontally at a speed V enters a region between two horizontally oriented plates of length L1. When the electrons reach a fluorescent screen located at a distance L2 past these plates, they have been deflected a vertical distance y from their original direction. If the speed of the electrons is doubled what is the new value of the deflection? A) y/4 B) 4y C) 2y D) y E) y/2

38. ACT A ring of negative, uniform charge density is placed on the xz-plane with the center of the ring at the origin. A positive charge moves along the y axis toward the center of the ring. At the moment the charge passes through the center of the ring ________ • A) its velocity and its acceleration reach their maximum values. • B) its velocity is maximum and its acceleration is zero. • C) its velocity and its acceleration have non-zero values but neither is at its maximum. • D) its velocity and its acceleration are both equal to zero. • E) its velocity is zero and its acceleration is maximum.

39. 16-5 The Electric Dipole in an External Electric Field 1. The motion of a dipole in an uniform external electric field

40. stable equilibrium unstable equilibrium A dipole in an uniform external electric field only rotates

41. 2.The energy of a dipole in an external electric field We choose

42. ACT The potential energies associated with four orientations of an electric dipole in an electric field are (1) -5U0, (2) -7U0, (3) 3U0, and (4) 5U0, where U0 is positive. Rank the orientations according to (a) the angle between the electric dipole moment    and the electric field   , and (b) the magnitude of the torque on the electric dipole, greatest first. (a) 4, 3, 1, 2 (b) 3, the 1 and 4 tie, then 2

43. ACT An electric dipole of dipole moment is placed in an electric fieldThe dipole is then slightly rotated a small angle θ away from its initial direction and released. When released the dipole will ________ A) stay in its new position, and angle θ away from its original position. B) continue to rotate in the direction of θ until its dipole moment is perpendicular to C) oscillate back an forth around the new position, an angle θ away from its original position. D) continue to rotate until its dipole moment is parallel to the and then oscillate around that position. E) go back to its original position.

44. _ + - - - - - 3. The motion of a dipole in an nonuniform external electric field A dipole placed in an nonuniform external electric field experiences not only a net force but also a torque. The motion is a combination of linear acceleration and rotation We can explain the phenomena that a charged rod can attracts small pieces of paper.

45. Act A neutral water molecule (H2O) in its vapor state has an electric dipole moment of magnitude (a)  How far apart are the molecule's centers of positive and negative charge (b)  If the molecule is placed in an electric field of , what maximum torque can the field exert on it? (Such a field can easily be set up in the laboratory.) (c)  How much work must an external agent do to turn this molecule end for end in this field, starting from its fully aligned position, for which q = 0?

46. Solution

47. Summary of Chapter 16 • Electric field is defined as the force per unit charge: • Force on a point charge q’ • Electric field lines are very useful for visualizing the electric field, as long as their limitations are taken into account.

48. Summary of Chapter 16, cont. • Electric field of a point charge: • Electric fields obey the superposition principle. • Electric dipole: equal and opposite charges separated by a distance L. The electric field is proportional to the dipole moment, which is:

49. Summary of Chapter16, cont. • An electric dipole in an external electric field feels a torque, and has potential energy: • Electric field due to a continuous charge distribution: