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ENE 325 Electromagnetic Fields and Waves

ENE 325 Electromagnetic Fields and Waves. Lecture 2 Static Electric Fields and Electric Flux density. Review (1). Vector quantity Magnitude Direction Coordinate systems Cartesian coordinates (x, y, z) Cylindrical coordinates (r,  , z) Spherical coordinates (r,  ,  ). Review (2).

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ENE 325 Electromagnetic Fields and Waves

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  1. ENE 325Electromagnetic Fields and Waves Lecture 2 Static Electric Fields and Electric Flux density

  2. Review (1) • Vector quantity • Magnitude • Direction • Coordinate systems • Cartesian coordinates (x, y, z) • Cylindrical coordinates (r, , z) • Spherical coordinates (r, , )

  3. Review (2) • Coulomb’s law • Coulomb’s force • electric field intensity (V/m)

  4. Review (3) • Key variables: • Coordinate system and its corresponding differential element • charge Q • a unit vector

  5. Outline • Electric field intensity in different charge configurations • infinite line charge • ring charge • surface charge • Examples from previous lecture • Electric flux density

  6. Infinite length line of charge • The derivation of and electric field at any point in space resulting from an infinite length line of charge. (good approximation)

  7. Infinite length line of charge • only varies with the radial distance  • select point P on  - z axis for convenience. • select a segment of charge dQ at distance –z, we then have

  8. Infinite length line of charge • Consider another segment at distance z, z components are cancelled out, we then have

  9. Infinite length line of charge • From We can write Total field

  10. Infinite length line of charge • Consider each segment • Ezcomponents are cancelled due to symmetry.

  11. Infinite length line of charge

  12. Ring of charge determine at (0,0,h) cancels each other

  13. Ring of charge Consider each segment:

  14. Surface charge • Surface charge density S (c/m2) dQ = Sdxdy Since this is an infinite place, Exand Ey components are cancelled due to symmetry.

  15. Surface charge • Consider each segment: Devide the whole area into infinite length of line charges Integrate over length y to get total electric field. Convert the radial component into cylindrical coordinates Ey components are cancelled out due to symmetry.

  16. Surface charge No dependence on a distance from the sheet

  17. Concentrate ring (alternative approach) for each ring Total field is integrated from  = 0 to  Then

  18. Volume charge • Volume charge density V (c/m3) • plasma • doped semiconductor • Complicate derivation due to so many differential elements and vectors.

  19. Ex1 Determine the distance between point P (5, /2, 10) and point Q (1, /3, 5) in cylindrical coordinates.

  20. Ex2 Determine a unit vector directed from (0, 0, h) to (r, , 0) in cylindrical coordinates.

  21. Ex3 Determine a unit vector from any point on z = -5 plane to the origin.

  22. Ex4 Find the area between 90135 on the surface of a sphere of a radius 1 m.

  23. Ex5 A charge Q1= 0.3C is located at (1,4,0). A charge Q2= 0.2 C is located at (3,0,0). Determine at point (0,0,5).

  24. Ex6 Determine at point (-2, -1, 4) given a line charge located at x = 2 and y = -4 with a charge density L = 20 nC/m.

  25. Ex7 Determine at the origin given a square sheet of charge located at z = -1 plane. The sheet is extended from -1  x  1 and -1  y  1with a surface charge density S = 2(x2+y2+1)3/2 nC/m2.

  26. Electric flux density • Negative charges are drawn to the outer sphere • Electric flux lines are radially directed away from inner sphere to outer sphere or begin from positive charges +Q and terminate on negative charges -Q.

  27. Electric flux density • Electric flux density, (C/m2) Note:  (chi) is a flux in Coulomb unit and is equal to charge Q on the sphere So we have where 0 = 8.854x10-12 Farad/m

  28. Electric flux density • The amount of flux passing through a surface isgiven by the product of and the amount of surface normal to. Same polarity charges repel one another Note: = surface vector • Dot product: for Cartesian coordinates. Dot product is a projection of A on B multiplies by B

  29. Electric flux density The flux through a surface that is an angle to the direction of flux a) is less than the flux through an equivalent surface normal to the direction of flux b) • In case the flux is varied over the surface,

  30. Ex8C/m2. Given the surface defined by  = 1 m, 090 and -1 z 1, calculate the flux through the surface.

  31. Ex9 A charge Q = 30 nC is located at the origin, determine the electric flux density at point (1, 2, -4) m.

  32. Ex10 Determine the flux through the area 1x1 mm2 on a surface of a cylinder at r = 10 m, z = 2 m,  = 53.2 given C/m2.

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