Understanding Maxwell's Equations: Electrostatics & Magnetostatics Overview

1. 3. Electrostatics Ruzelita Ngadiran

2. Chapter 4 Overview

3. Maxwell’s equations Maxwell’s equations: Where; E = electric field intensity D = electric flux density ρv = electric charge density per unit volume H = magnetic field intensity B = magnetic flux density

4. Maxwell’s Equations God said: And there was light!

5. Maxwell’s equations Maxwell’s equations: Relationship: D = ε E B = µ H ε = electrical permittivity of the material µ = magnetic permeability of the material

6. Maxwell’s equations • For staticcase, ∂/∂t = 0. • Maxwell’s equations is reduced to: ElectrostaticsMagnetostatics

7. Charge and current distributions • Charge may be distributed over a volume, a surface or a line. • Electric field due to continuous charge distributions:

8. Charge and current distributions • Volume charge density, ρv is defined as: • Total charge Q • contained in • a volume V is:

9. Charge and current distributions • Surface charge density • Total charge Q on a surface:

10. Charge and current distributions • Line charge density • Total charge Q along a line

11. Charge Distributions Volume charge density: Total Charge in a Volume Surface and Line Charge Densities

12. Example 1 Calculate the total charge Q contained in a cylindrical tube of charge oriented along the z-axis. The line charge density is , where zis the distance in meters from the bottom end of the tube. The tube length is 10 cm.

13. Solution to Example 1 The total charge Q is:

14. Example 2 Find the total charge over the volume with volume charge density:

15. Solution to Example 2 The total charge Q:

16. Current Density For a surface with any orientation: J is called the current density

17. Convection vs. Conduction

18. Coulomb’s Law Electric field at point P due to single charge Electric force on a test charge placed at P Electric flux density D

19. For acting on a charge • For a material with electrical permittivity, ε: D = εE where: ε = εR ε0 ε0 = 8.85 × 10−12≈ (1/36π) × 10−9 (F/m) • For most material and under most condition, ε is constant, independent of the magnitude and direction of E

20. E-field due to multipoint charges • At point P, the electric field E1 due to q1 alone: • At point P, the electric field E1 due to q2 alone:

21. Electric Field Due to 2 Charges

22. Example 3 Two point charges with and are located in free space at (1, 3,−1) and (−3, 1,−2), respectively in a Cartesian coordinate system. Find: (a) the electric field E at (3, 1,−2) (b) the force on a 8 × 10−5 C charge located at that point. All distances are in meters.

23. Solution to Example 3 • The electric field E with ε = ε0 (free space) is given by: • The vectors are:

24. Solution to Example 3 • a) Hence, • b) We have

25. Electric Field Due to Charge Distributions Field due to:

26. Cont.

28. Example 4 Find the electric field at a point P(0, 0, h) in free space at a height h on the z-axis due to a circular disk of charge in the x–y plane with uniform charge density ρs as shown. Then evaluate E for the infinite-sheet case by letting a→∞.

29. Solution to Example 4 • A ring of radius r and width dr has an area ds = 2πrdr • The charge is: • The field due to the ring is:

30. Solution to Example 4 • The total electric field at P is With plus sign corresponds to h>0, minus sign corresponds to h<0. • For an infinite sheet of charge with a =∞,

31. Gauss’s Law Application of the divergence theorem gives:

32. Example 5 • Use Gauss’s law to obtain an expression for E in free space due to an infinitely long line of charge with uniform charge density ρlalong the z-axis.

33. Solution to Example 5 Construct a cylindrical Gaussian surface. The integral is: Equating both equations, and re-arrange, we get:

34. Solution to Example 5 Then, use , we get: Note: unit vector is inserted for E due to the fact that E is a vector in direction.

35. Applying Gauss’s Law Construct an imaginary Gaussian cylinder of radius r and height h:

36. Electric Scalar Potential Minimum force needed to move charge against E field:

37. Electric Scalar Potential

38. Electric Potential Due to Charges For a point charge, V at range R is: In electric circuits, we usually select a convenient node that we call ground and assign it zero reference voltage. In free space and material media, we choose infinity as reference with V = 0. Hence, at a point P For continuous charge distributions:

39. Relating E to V

40. Cont.

41. (cont.)

42. Poisson’s & Laplace’s Equations In the absence of charges:

43. Conductivity • Conductivity – characterizes the ease with which charges can move freely in a material. • Perfect dielectric, σ = 0. Charges do not move inside the material • Perfect conductor, σ = ∞. Charges move freely throughout the material

44. Conductivity • Drift velocity of electrons, in a conducting material is in the opposite direction to the externally applied electric field E: • Hole drift velocity, is in the same direction as the applied electric field E: where: µe = electron mobility (m2/V.s) µh= hole mobility (m2/V.s)

45. Conductivity • Conductivity of a material, σ, is defined as: where ρve = volume charge density of free electrons ρvh = volume charge density of free holes Ne = number of free electrons per unit volume Nh = number of free holes per unit volume e = absolute charge = 1.6 × 10−19 (C)

46. Conductivity • Conductivities of different materials:

47. Conductivity ve = volume charge density of electrons he = volume charge density of holes e = electron mobility h = hole mobility Ne = number of electrons per unit volume Nh = number of holes per unit volume

48. Conduction Current Conduction current density: Note how wide the range is, over 24 orders of magnitude

50. Resistance Longitudinal Resistor For any conductor: