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Electrostatic fields

Electrostatic fields. Sandra Cruz-Pol, Ph. D. INEL 4151 ECE UPRM Mayagüez, PR. Some applications. Power transmission, X rays, lightning protection Solid-state Electronics: resistors, capacitors, FET Computer peripherals: touch pads, LCD, CRT

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Electrostatic fields

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

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

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

  4. Point charges Coulomb’s Law (1785) • Force one charge exerts on another where k= 9 x 109 or k = 1/4peo + R + *Superposition applies

  5. Force with direction

  6. Example • Example: Point charges 5nC and -2nC are located at r1=(2,0,4) and r2=(-3,0,5), respectively. • Find the force on a 1nC point charge, Qx, located at (1,-3,7) • Apply superposition:

  7. Electric field intensity • 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 rx=(1,-3,7).

  8. If we have many charges

  9. The total E-field intensity is

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

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

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

  13. Find E from Surface charge • Sheet of charge w/uniform density rS z y

  14. SURFACE charge • Due to SYMMETRY the r component cancels out.

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

  16. Find E from Volume charge • sphere of charge w/uniform density, rv P(0,0,z) dE a (Eq. *) (r’,q’,f’) q’ Differentiating (Eq. *) rv f’ x

  17. Find E from Volume charge • Substituting… P(0,0,z) dE (r’,q’,f’) q’ rv f’ x 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).

  19. z Cont… sheet of charge y=2 x=2 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. Gauss’s Law • The total electric flux Y, through any closed surface is equal to the total charge enclosed by that surface.

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

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

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

  26. P.E. 4.7 A point charge of 30nC is located at the origin, while plane y=3 carries charge 10nC/m2. Find D at (0, 4, 3)

  27. P.E. 4.8 If C/m2 . Find : • volume charge density at (-1,0,3) • Flux thru the cube defined by • 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) • Gauss’s Law (symmetrical cases) • Electric Potential (uses scalar, not vectors)

  30. Electric Potential, V • The work done to move a charge Q from A to B is • The (-) means the work is done by an external force. • The total work= potential energy required in moving Q: • 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 rk: (apply superposition) For Line Charges: For Surface charges: For Volume charges:

  32. P.E. 4.10 A point charge of -4mC is located at (2,-1,3) A point charge of 5mC is located at (0,4,-2) A point charge of 3mC 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) VA=+4.8V r0=|(0,0,0)-(10,20,0)|=22.36 and rA=|(3,0,5)-(10,20,0)|= 21.2

  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 VAB

  35. Relation between E and V V is independent of the path taken. 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. A

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

  37. P.E. 4.12 a) (0,5,0)→(2,5,0) →(2,-1,0) Given that E=(3x2+y)ax +x ay kV/m, find the work done in moving a -2mC charge from (0,5,0) to (2,-1,0) by taking the straight-line path. 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. z P r1 For far away observation points (r>>d): Q+ r r2 d y Q-

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

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