UPB / ETTI O.DROSU Electrical Engineering 2

1. UPB / ETTIO.DROSUElectrical Engineering 2 Lecture 3: Electrostatics: Electrostatic Potential; Charge Dipole; Visualization of Electric Fields; Potentials; Gauss’s Law and Applications; Conductors and Conduction Current 1

2. Objectives • To continue our study of electrostatics with • electrostatic potential; • charge dipole; • visualization of electric fields and potentials; • Gauss’s law and applications; • conductors and conduction current. 2

3. P Q Electrostatic Potential of a Point Charge at the Origin spherically symmetric 3

4. Electrostatic Potential Resulting from Multiple Point Charges Q2 P(R,q,f) Q1 O No longer spherically symmetric! 4

5. Electrostatic Potential Resulting from Continuous Charge Distributions  line charge  surface charge  volume charge 5

6. -Q +Q d Charge Dipole • An electric charge dipole consists of a pair of equal and opposite point charges separated by a small distance (i.e., much smaller than the distance at which we observe the resulting field). 6

7. +Q -Q Dipole Moment • Dipole momentp is a measure of the strength • of the dipole and indicates its direction p isin the direction from the negative point charge to the positive point charge 7

8. P observation point z +Q d/2 d/2 q -Q Electrostatic Potential Due to Charge Dipole 8

9. Electrostatic Potential Due to Charge Dipole cylindrical symmetry 9

10. Electrostatic Potential Due to Charge Dipole P q d/2 d/2 10

11. Electrostatic Potential Due to Charge Dipole in the Far-Field • assume R>>d • zeroth order approximation: not good enough! 11

12. Electrostatic Potential Due to Charge Dipole in the Far-Field • first order approximation from geometry: q d/2 d/2 lines approximately parallel 12

13. Electrostatic Potential Due to Charge Dipole in the Far-Field • Taylor series approximation: 13

14. Electrostatic Potential Due to Charge Dipole in the Far-Field • In terms of the dipole moment: 14

15. Electric Field of Charge Dipole in the Far-Field 15

16. Visualization of Electric Fields • An electric field (like any vector field) can be visualized using flux lines (also called streamlinesor lines of force). • A flux line is drawn such that it is everywhere tangent to the electric field. • A quiver plot is a plot of the field lines constructed by making a grid of points. An arrow whose tail is connected to the point indicates the direction and magnitude of the field at that point. 16

17. Visualization of Electric Potentials • The scalar electric potential can be visualized using equipotential surfaces. • An equipotential surface is a surface over which V is a constant. • Because the electric field is the negative of the gradient of the electric scalar potential, the electric field lines are everywhere normal to the equipotential surfaces and point in the direction of decreasing potential. 17

18. Visualization of Electric Fields • Flux lines are suggestive of the flow of some fluid emanating from positive charges (source) and terminating at negative charges (sink). • Although electric field lines do NOT represent fluid flow, it is useful to think of them as describing the flux of something that, like fluid flow, is conserved. 18

19. charged sphere (+Q) + + + metal + insulator Faraday’s Experiment 19

20. Faraday’s Experiment • Two concentric conducting spheres are separated by an insulating material. • The inner sphere is charged to +Q. Theouter sphere is initially uncharged. • The outer sphere is groundedmomentarily. • The charge on the outer sphere is found to be -Q. 20

21. Faraday’s Experiment • Faraday concluded there was a “displacement” from the charge on the inner sphere through the inner sphere through the insulator to the outer sphere. • The electric displacement (or electric flux) is equal in magnitude to the charge that produces it, independent of the insulating material and the size of the spheres. 21

22. +Q -Q Electric Displacement (Electric Flux) 22

23. Electric (Displacement) Flux Density • The density of electric displacement is the electric (displacement) flux density, D. • In free space the relationship between flux density and electric field is 23

24. Electric (Displacement) Flux Density • The electric (displacement) flux density for a point charge centered at the origin is 24

25. Gauss’s Law • Gauss’s law states that “the net electric flux emanating from a close surface S is equal to the total charge contained within the volume V bounded by that surface.” 25

26. S ds V Gauss’s Law By convention, ds is taken to be outward from the volume V. Since volume charge density is the most general, we can always write Qencl in this way. 26

27. Applications of Gauss’s Law • Gauss’s law is an integral equation for the unknown electric flux density resulting from a given charge distribution. known unknown 27

28. Applications of Gauss’s Law • In general, solutions to integral equations must be obtained using numerical techniques. • However, for certain symmetric charge distributions closed form solutions to Gauss’s law can be obtained. 28

29. Applications of Gauss’s Law • Closed form solution to Gauss’s law relies on our ability to construct a suitable family of Gaussian surfaces. • A Gaussian surface is a surface to which the electric flux density is normal and over which equal to a constant value. 29

30. Q Electric Flux Density of a Point Charge Using Gauss’s Law Consider a point charge at the origin: 30

31. Electric Flux Density of a Point Charge Using Gauss’s Law (1) Assume from symmetry the form of the field (2) Construct a family of Gaussian surfaces spherical symmetry spheres of radius r where 31

32. Electric Flux Density of a Point Charge Using Gauss’s Law (3) Evaluate the total charge within the volume enclosed by each Gaussian surface 32

33. Q Electric Flux Density of a Point Charge Using Gauss’s Law Gaussian surface R 33

34. Electric Flux Density of a Point Charge Using Gauss’s Law (4) For each Gaussian surface, evaluate the integral surface area of Gaussian surface. magnitude of D on Gaussian surface. 34

35. Electric Flux Density of a Point Charge Using Gauss’s Law (5) Solve for D on each Gaussian surface 35

36. a b Electric Flux Density of a Spherical Shell of Charge Using Gauss’s Law Consider a spherical shell of uniform charge density: 36

37. Electric Flux Density of a Spherical Shell of Charge Using Gauss’s Law (1) Assume from symmetry the form of the field (2) Construct a family of Gaussian surfaces spheres of radius r where 37

38. a b Electric Flux Density of a Spherical Shell of Charge Using Gauss’s Law • Here, we shall need to treat separately 3 sub-families of Gaussian surfaces: 1) 2) 3) 38

39. Electric Flux Density of a Spherical Shell of Charge Using Gauss’s Law Gaussian surfaces for which Gaussian surfaces for which Gaussian surfaces for which 39

40. Electric Flux Density of a Spherical Shell of Charge Using Gauss’s Law (3) Evaluate the total charge within the volume enclosed by each Gaussian surface 40

41. Electric Flux Density of a Spherical Shell of Charge Using Gauss’s Law • For • For 41

42. Electric Flux Density of a Spherical Shell of Charge Using Gauss’s Law • For 42

43. Electric Flux Density of a Spherical Shell of Charge Using Gauss’s Law (4) For each Gaussian surface, evaluate the integral surface area of Gaussian surface. magnitude of D on Gaussian surface. 43

44. Electric Flux Density of a Spherical Shell of Charge Using Gauss’s Law (5) Solve for D on each Gaussian surface 44

45. Electric Flux Density of a Spherical Shell of Charge Using Gauss’s Law 45

46. Electric Flux Density of a Spherical Shell of Charge Using Gauss’s Law • Notice that for r > b Total charge contained in spherical shell 46

47. 0.7 0.6 0.5 0.4 (C/m) r D 0.3 0.2 0.1 0 0 1 2 3 4 5 6 7 8 9 10 R Electric Flux Density of a Spherical Shell of Charge Using Gauss’s Law 47

48. Electric Flux Density of an Infinite Line Charge Using Gauss’s Law Consider a infinite line charge carrying charge per unit length of qel: z 48

49. Electric Flux Density of an Infinite Line Charge Using Gauss’s Law (1) Assume from symmetry the form of the field (2) Construct a family of Gaussian surfaces cylinders of radius r where 49

50. Electric Flux Density of an Infinite Line Charge Using Gauss’s Law (3) Evaluate the total charge within the volume enclosed by each Gaussian surface cylinder is infinitely long! 50