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

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

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

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

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

6. 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. 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. 8 Infinite length line of charge Consider another segment at distance z, z components are cancelled out, we then have

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

10. 10 Infinite length line of charge Consider each segment E z components are cancelled due to symmetry.

11. 11 Infinite length line of charge

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

13. 13 Ring of charge Consider each segment:

14. 14 Surface charge Surface charge density  S (c/m 2 ) dQ =  S dxdy Since this is an infinite place, E x and E y components are cancelled due to symmetry.

15. 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 E y components are cancelled out due to symmetry.

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

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

18. 18 Volume charge Volume charge density  V (c/m 3 ) plasma doped semiconductor Complicate derivation due to so many differential elements and vectors.

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

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

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

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

23. 23 Ex5 A charge Q 1 = 0.3  C is located at (1,4,0). A charge Q 2 = 0.2  C is located at (3,0,0). Determine at point (0,0,5).

24. 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. 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(x 2 +y 2 +1) 3/2 nC/m 2.

26. 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. 27 Electric flux density Electric flux density, (C/m 2 ) Electric flux density, (C/m 2 ) 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. 28 The amount of flux passing through a surface is given 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 Electric flux density

29. 29 In case the flux is varied over the surface, 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)

30. 30 Ex8 C/m 2. Given the surface defined by  = 1 m, 0    90  and -1  z  1, calculate the flux through the surface.

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

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