1. Electrostatic fields Sandra Cruz-Pol, Ph. D. INEL 4151 ECE UPRM Mayagüez, PR

2. Some applications Power transmission, X rays, lightning protection Power transmission, X rays, lightning protection Solid-state Electronics: resistors, capacitors, FET Solid-state Electronics: resistors, capacitors, FET Computer peripherals: touch pads, LCD, CRT Computer peripherals: touch pads, LCD, CRT Medicine: electrocardiograms, electroencephalograms, monitoring eye activity Medicine: electrocardiograms, electroencephalograms, monitoring eye activity Agriculture: seed sorting, moisture content monitoring, spinning cotton, … Agriculture: seed sorting, moisture content monitoring, spinning cotton, … Art: spray painting Art: spray painting …

3. We will study Electric charges: Coulomb's Law Coulomb's Law Gauss’s Law Gauss’s Law

4. Coulomb’s Law (1785) Force one charge exerts on another Force one charge exerts on another where k= 9 x 10 9 or k = 1/4  o + + R Point charges *Superposition applies

5. Force with direction

6. Example Example: Point charges 5nC and -2nC are located at r 1 =(2,0,4) and r 2 = (-3,0,5), respectively. a)Find the force on a 1nC point charge, Q x, located at (1,-3,7) Apply superposition:

7. Electric field intensity Is the force per unit charge when placed in the E field Is the force per unit charge when placed in the E field Example: Point charges 5nC and - 2nC are located at (2,0,4) and (- 3,0,5), respectively. b) Find the E field at r x =(1,-3,7).

8. If we have many charges Line charge density,  L C/m Surface charge density  S C/m 2 Volume charge density  v C/m 3

9. The total E-field intensity is

10. Find E from LINE charge Line charge w/uniform charge density,  L Line charge w/uniform charge density,  L x z B A R dl dE  0 T (0,0,z’) (x,y,z)

11. LINE charge Substituting in: Substituting in: x z B A R dl dE  0 T (0,0,z’) (x,y,z)

12. More Charge distributions Point charge Point charge Line charge Line charge Surface charge Surface charge Volume charge Volume charge

13. Find E from Surface charge Sheet of charge w/uniform density  S Sheet of charge w/uniform density  S y z

14. SURFACE charge Due to SYMMETRY Due to SYMMETRY the  component cancels out. the  component cancels out.

15. More Charge distributions Point charge Point charge Line charge Line charge Surface charge Surface charge Volume charge Volume charge

16. Find E from Volume charge sphere of charge w/uniform density,  v sphere of charge w/uniform density,  v x ’’ ’’ ( r ’,  ’  ’  P( 0,0,z) vv dE (Eq. *)  Differentiating (Eq. *)

17. Find E from Volume charge Substituting… Substituting… x ’’ ’’ ( r ’,  ’  ’  P( 0,0,z) vv dE De donde salen los limites de R?

18. P.E. 4.5 A square plate at plane z=0 and carries a charge mC/m2. Find the total charge on the plate and the electric field intensity at (0,0,10). A square plate at plane z=0 and carries a charge mC/m2. Find the total charge on the plate and the electric field intensity at (0,0,10).

19. Cont… sheet of charge x=2 y=2 z Due to symmetry only Ez survives:

20. Electric Flux Density D is independent of the medium in which the charge is placed.

21. Gauss’s Law

22. The total electric flux , through any closed surface is equal to the total charge enclosed by that surface. The total electric flux , through any closed surface is equal to the total charge enclosed by that surface.

23. Some examples: Finding D at point P from the charges: Point Charge is at the origin. Point Charge is at the origin. Choose a spherical dS Choose a spherical dS Note where D is perpendicular to this surface. Note where D is perpendicular to this surface. D P r charge

24. Some examples: Finding D at point P from the charges: Infinite Line Charge Infinite Line Charge Choose a cylindrical dS Choose a cylindrical dS Note that integral =0 at top and bottom surfaces of cylinder Note that integral =0 at top and bottom surfaces of cylinder D  P Line charge

25. Some examples: Find D at point P from the charges: Infinite Sheet of charge Infinite Sheet of charge Choose a cylindrical box cutting the sheet Choose a cylindrical box cutting the sheet Note that D is parallel to the sides of the box. sheet of charge D D Area A

26. P.E. 4.7 A point charge of 30nC is located at the origin, while plane y=3 carries charge 10nC/m 2. Find D at (0, 4, 3) Find D at (0, 4, 3)

27. P.E. 4.8 If C/m2. Find : volume charge density at (-1,0,3) volume charge density at (-1,0,3) Flux thru the cube defined by Flux thru the cube defined by Total charge enclosed by the cube Total charge enclosed by the cube

28. Review Point charge or volume Charge distribution Line charge distribution Sheet charge distribution

29. We will study Electric charges: Coulomb's Law (general cases) Coulomb's Law (general cases) Gauss’s Law (symmetrical cases) Gauss’s Law (symmetrical cases) Electric Potential (uses scalar, not vectors) Electric Potential (uses scalar, not vectors)

30. Electric Potential, V The work done to move a charge Q from A to B is The work done to move a charge Q from A to B is The (-) means the work is done by an external force. The (-) means the work is done by an external force. The total work= potential energy required in moving Q: The total work= potential energy required in moving Q: The energy per unit charge= potential difference between the 2 points: The energy per unit charge= potential difference between the 2 points: V is independent of the path taken.

31. The Potential at any point is the potential difference between that point and a chosen reference point at which the potential is zero. (choosing infinity): For many Point charges at r k : For many Point charges at r k : (apply superposition) For Line Charges: For Surface charges: For Volume charges: For Volume charges:

32. P.E. 4.10 A point charge of -4  C is located at (2,-1,3) A point charge of 5  C is located at (0,4,-2) A point charge of 3  C is located at the origin Assume V(∞)=0 and Find the potential at (-1, 5, 2) =10.23 kV

33. Example A line charge of 5nC/m is located on line x=10, y=20 Assume V(0,0,0)=0 and Find the potential at A(3, 0, 5)  0 =|(0,0,0)-(10,20,0)|=22.36 and  A =|(3,0,5)-(10,20,0)|= 21.2 V A =+4.8V

34. P.E. 4.11 A point charge of 5nC is located at the origin V(0,6,-8)=2V and Find the potential at A(-3, 2, 6) Find the potential at B(1,5,7), the potential difference V AB

35. Relation between E and V A B Esto aplica sólo a campos estáticos. Significa que no hay trabajo NETO en mover una carga en un paso cerrado donde haya un campo estático E. V is independent of the path taken.

36. Static E satisfies: A B Condition for Conservative field = independent of path of integration

37. P.E. 4.12 Given that E=(3x 2 +y)a x +x a y kV/m, find the work done in moving a -2  C charge from (0,5,0) to (2,-1,0) by taking the straight-line path. a) (0,5,0)→(2,5,0) →(2,-1,0) b) y = 5-3x

38. Example Given the potential Find D at. In spherical coordinates:

39. Electric Dipole Is formed when 2 point charges of equal but opposite sign are separated by a small distance. Is formed when 2 point charges of equal but opposite sign are separated by a small distance. P y r1r1 r2r2 r z d Q+ Q- For far away observation points ( r>>d ):

40. Energy Density in Electrostatic fields It can be shown that the total electric work done is: It can be shown that the total electric work done is: