Electric Fields. Chapter 26 The Electric Field. Chapter 26 Goal: To learn how to calculate and use the electric field. Slide 26-2

Transcription

1 Chapter 26 The Electric Field Electric Fields Chapter 26 Goal: To learn how to calculate and use the electric field. Slide 26-2

2 Electric Field Shows how the field acts on a POSTIVE test charge! E = Force/unit Charge

3 E: The Electric Field Lines Shows how the Force acts on a STATIC Positive Test Charge.

4 Electric Field Definition: F F qe E q Electric field is the force per unit coulomb: it is the force that +1C would feel if it were placed at this location. E field is in N/C. E field is a VECTOR. E field shows how the force acts on a POSTIIVE test charge! kq Electric Field E due to q : E 1 1 r 2

5 Electric Field Strengths

6 Electric Field Lines: Direction is for a POSITIVE Test charge direction of E-field at any point is tangent to the field line magnitude of E-field at any point is proportional to line density number of lines leaving or entering a charged object is proportional to the charge on the object Electric Force is in the direction of the E field for positive charges and in the opposite direction for negative charges.

7 Two Positive Charges Electric Field Fig 23-23a, p.725

8 Electric Field Visualization

9 MIT Shockwave Visual EM isualizations/guidedtour/tour.htm

10 Equal Distance Equal Force A B C

11 WARNING: The Electric Force and Field Directions not always the same! Depends on if the charge put into the field is positive or negative. The Electric Force is in the direction of the E field for positive charges and in the opposite direction for negative charges! Section 23.4

12 Electric Field Vector The electric field at P due to q. To find the sign and directions ask, how would the charge act on a POSITIVE test charge? Repulsive: Positive E p k r q r ˆ 2 Attractive: Negative

13 Superposition of Electric Field Vectors E E E P 1P 2P What is the net field at P due to all charges? Problem Solving Strategy 1. Draw FBD 2. Calculate Field Magnitudes 3. Resolve into components 4. Add components 5. Express as ijk vector 6. Find magnitude and direction of resultant field. Test extrema! Sketch field lines!

14 Electric Dipole: Bisector Electric Dipole: Two point charges of equal but opposite magnitude are lying on the y-axis. Ignore gravity. They are equidistant from the point P, which lies on the x-axis. What is the net electric field at P? Draw the field vectors showing the net field. Derive a general solution. What is the electric field for a point much further away than the charge separation distance? q 4.00 C, q 4.00 C 1 2

15 Dipole Electric Field Bisector Near: E y 2kQa ( x a ) 2 2 3/2 Far: E y 2kQa 3 x E In practice, we almost always observe the electric field of a dipole only for distance much greater than the charge separation and the E ~1/r 3 holds everywhere.

16 The Dipole Electric Field at Two Points 2013 Pearson Education, Inc. Slide 26-32

17 Electric Dipoles Two equal but opposite charges separated by a small distance form an electric dipole. The figure shows two examples Pearson Education, Inc. Slide 26-30

18 The Dipole Moment It is useful to define the dipole moment p, shown in the figure, as the vector: The SI units of the dipole moment are C m Pearson Education, Inc. Slide 26-31

19 The Electric Field of a Dipole The electric field at a point on the axis of a dipole is: where r is the distance measured from the center of the dipole. The electric field in the plane that bisects and is perpendicular to the dipole is This field is opposite to the dipole direction, and it is only half the strength of the on-axis field at the same distance Pearson Education, Inc. Slide 26-33

20 Electric Dipole: H20 Although an electric dipole has zero net charge, it does have a net electric field. Some molecules, such as the water molecule, have permanent electric dipoles. The polar nature of water molecules allows them to bond to each other in groups and is associated with the high surface tension of water. The dipole moment of water provides a "handle" for interaction with microwave electric fields in a microwave oven. More on electric dipoles next week

21 Example 26.2 The Electric Field of a Water Molecule 2013 Pearson Education, Inc. Slide 26-34

22 BTW: You might see the Dipole with this orientation: SAME RESULT!

23 This Form of Coulomb s Law for the Electric Field is only true for point charges and systems of point charges! E p k r q r ˆ 2 E p E i i

24 Continuous Charge Distribution Note: For spheres, cylinders, infinite planes, rods, etc with high symmetry, the electric field is found using Gauss s Law which we ll do next chapter! dq E de k r r 2 ˆ Q L Q A Q V linear charge density surface charge density volume charge density dq dq dq dx da dv

25 Recall from Physics 40

26 Finite Uniform Rod: Bisector A thin rod of length l and uniform charge per unit length λ lies along the x axis, as shown. Find the electric field at P, a distance y from the rod along its perpendicular bisector, and the electric field at P of the rod if it is of infinite length. USE SYMMETRY ARGUMENTS TO SIMPLIFY THE SOLUTION!!!!

27 Finite Uniform Rod: Bisector E y kq 2 2 y y ( l / 2) If y>>l E y kq y 2 Far away from the rod, the field looks like that of a point charge.

28 kq y Far Away from the Finite Line: Ey 2

29 Infinite Line of Charge Bisecting the Rod kq 2kQ k2 lim Ex lim lim lim 2k L L 2 2 L 2 L y y ( L / 2) yl ( y / L) 1 y y Ex 2k y Notice: The electric field of an infinitely long charged rod decreases as 1/r! This is a good approximation for a finite line of charge points far from the ends.

30 An infinite line of charge: We will find the same result using Gauss s Law to find the field with much greater ease in chapter 27. The rod is VERY Important because it models a wire!! E 2 2 k r r 4 r 2 r 0 0 E r 2 r 0

31 An Infinite Line of Charge The electric field of a thin, uniformly charged rod may be written: If we now let L, the last term becomes simply 1 and we re left with: 2013 Pearson Education, Inc. Slide 26-54

32 Check Out Hyperphysics

33 How would you set this up? Fig , p. 672

34 Next time: NonUniform Charge Distribution

35 A Ring of Charge Consider the on-axis electric field of a positively charged ring of radius R. Define the z-axis to be the axis of the ring. The electric field on the z-axis points away from the center of the ring, increasing in strength until reaching a maximum when z R, then decreasing: 2013 Pearson Education, Inc. Slide 26-55

36 2013 Pearson Education, Inc. Semi Circle

37 Ring of Charge! E ring kxq x 2 R 2 3/2

38 Finite Thin Disc! To find the electric field at P, use the result from the ring as differential element and integrate r from zero to R: kxq Ering 2 2 x R 3/2 E Disk 2 k 1 x x R Pearson Education, Inc.

39 Infinite Charged Plane x lim EDisk lim 2 k 1 2 k R R 2 2 x R E 2 2 k E 2 0 Infinite Charged Plane has a constant Uniform Field EVERYWHERE!!!!

40 Infinite Charged Plane E 2 0

41 Rank in order, from largest to smallest, the electric field strengths E a to E e at these five points near an infinite plane of charge. 1. E a = E b = E c = E d = E e 2. E a > E c > E b > E e > E d 3. E b = E c = E d = E e > E a 4. E a > E b = E c > E d = E e 5. E e > E d > E c > E b > E a

42 Rank in order, from largest to smallest, the electric field strengths E a to E e at these five points near an infinite plane of charge. 1. E a = E b = E c = E d = E e 2. E a > E c > E b > E e > E d 3. E b = E c = E d = E e > E a 4. E a > E b = E c > E d = E e 5. E e > E d > E c > E b > E a

43 Parallel Plates: In the center we assume they are infinite and that the field is constant and Uniform and twice what it is for one plate! E 0

44 Uniform Infinite Parallel Plates E 0 IMPORTANT!!!! The electric field is uniform, therefore the force has equal magnitude at any distance from the plates. Therefore, the acceleration is constant between the plates. This is the model for parallel plate capacitors that are an important device for storing electric energy! Fig 23-25, p.726

45 The Parallel-Plate Capacitor The figure shows two electrodes, one with charge Q and the other with Q placed face-toface a distance d apart. This arrangement of two electrodes, charged equally but oppositely, is called a parallel-plate capacitor. Capacitors play important roles in many electric circuits. Slide 26-64

46 The Parallel-Plate Capacitor The figure shows two capacitor plates, seen from the side. Because opposite charges attract, all of the charge is on the inner surfaces of the two plates. Inside the capacitor, the net field points toward the negative plate. Outside the capacitor, the net field is zero. Slide 26-65

47 Example 26.7 Charge Density on a Cell Wall Slide 26-76

48 Parallel Plate Capacitor Two 6.0 cm diameter circular electrodes are spaced 5.0 mm apart. The capacitor is charged by transferring electrons from the right to the left electrode. An electron is released from rest at the surface of the negative electrode. How long does it take the electron to cross to the positive electrode? What speed does it collide? Assume the space between the plates is a vacuum.

49 Proton Gun

50 Cathode Ray Tube

51 An electron enters the lower left side of a parallel plate capacitor and exits at the upper right side as shown. The initial speed of the electron is vi. Assume that the electric field is uniform everywhere between the plates. Find the magnitude of the electric field. 6 vi 7.00x10 m/ s

52 Dipoles in a Uniform Electric Field The figure shows an electric dipole placed in a uniform external electric field. The net force on the dipole is zero. The electric field exerts a torque on the dipole which causes it to rotate. Slide 26-86

53 Dipoles in a Uniform Electric Field The figure shows an electric dipole placed in a uniform external electric field. The torque causes the dipole to rotate until it is aligned with the electric field, as shown. Notice that the positive end of the dipole is in the direction in which E points. Slide 26-87

54 QuickCheck Which dipole experiences no net force in the electric field? A. A. Dipole A. B. Dipole B. C. Dipole C. D. Both dipoles A and C. E. All three dipoles. B. C. Slide 26-88

55 QuickCheck Which dipole experiences no net force in the electric field? A. A. Dipole A. B. Dipole B. C. Dipole C. D. Both dipoles A and C. E. All three dipoles. B. C. Slide 26-89

56 QuickCheck Which dipole experiences no net torque in the electric field? A. A. Dipole A. B. Dipole B. C. Dipole C. D. Both dipoles A and C. E. All three dipoles. B. C. Slide 26-90

57 QuickCheck Which dipole experiences no net torque in the electric field? A. A. Dipole A. B. Dipole B. C. Dipole C. D. Both dipoles A and C. E. All three dipoles. B. C. Slide 26-91

58 Dipoles in a Uniform Electric Field The figure shows a sample of permanent dipoles, such as water molecules, in an external electric field. All the dipoles rotate until they are aligned with the electric field. This is the mechanism by which the sample becomes polarized. Slide 26-92

59 The Torque on a Dipole The torque on a dipole placed in a uniform external electric field is Slide 26-93

60 Example The Angular Acceleration of a Dipole Dumbbell Slide 26-94

64 QuickCheck A proton is moving to the right in a vertical electric field. A very short time later, the proton s velocity is Slide 26-82

65 QuickCheck A proton is moving to the right in a vertical electric field. A very short time later, the proton s velocity is Slide 26-83

66 QuickCheck 26.9 Two protons, A and B, are next to an infinite plane of positive charge. Proton B is twice as far from the plane as proton A. Which proton has the larger acceleration? A. Proton A. B. Proton B. C. Both have the same acceleration. Slide 26-61

67 QuickCheck 26.9 Two protons, A and B, are next to an infinite plane of positive charge. Proton B is twice as far from the plane as proton A. Which proton has the larger acceleration? A. Proton A. B. Proton B. C. Both have the same acceleration. Slide 26-62

68 QuickCheck Which electric field is responsible for the proton s trajectory? A. B. C. D. E. Slide 26-84

69 QuickCheck Which electric field is responsible for the proton s trajectory? A. B. C. D. E. Slide 26-85

70 QuickCheck 26.4 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. Slide 26-39

71 QuickCheck 26.4 Two protons, A and B, are in an electric field. Which proton has the larger acceleration? Stronger field where field lines are closer together. A. Proton A. B. Proton B. C. Both have the same acceleration. Weaker field where field lines are farther apart. Slide 26-40

72 QuickCheck 26.8 At the dot, the y-component of the electric field due to the shaded region of charge is A. B. C. D. E. Slide 26-52

73 QuickCheck 26.8 At the dot, the y-component of the electric field due to the shaded region of charge is A. B. C. D. E. Slide 26-53