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 static case, ∂/∂t = 0.
Maxwell’s equations is reduced to:
Electrostatics Magnetostatics

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 z is 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 = ε E
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

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 ρl along 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:

51. G’=0 if the insulating material is air or a perfect dielectric with zero conductivity.

52. Joule’s Law
The power dissipated in a volume containing electric field E and current density J is:
For a coaxial cable:
For a resistor, Joule’s law reduces to:

53. Piezoresistivity The Greek word piezein means to press
R0 = resistance when F = 0
F = applied force
A0 = cross-section when F = 0
 = piezoresistive coefficient of material

54. Piezoresistors

55. Wheatstone Bridge
Wheatstone bridge is a high sensitivity circuit for measuring small changes in resistance

56. Dielectric Materials

57. Polarization Field
P = electric flux density induced by E

58. Electric Breakdown
Electric Breakdown

59. Boundary Conditions

60. Summary of Boundary Conditions
Remember E = 0 in a good conductor

61. Conductors
Net electric field inside a conductor is zero

62. Field Lines at Conductor Boundary
At conductor boundary, E field direction is always perpendicular to conductor surface

63. Capacitance

64. Capacitance
For any two-conductor configuration:
For any resistor:

66. Application of Gauss’s law gives:
Q is total charge on inside of outer cylinder, and –Q is on outside surface of inner cylinder

67. Tech Brief 8: Supercapacitors
For a traditional parallel-plate capacitor, what is the maximum attainable energy density?
Mica has one of the highest dielectric strengths ~2 x 10**8 V/m.
If we select a voltage rating of 1 V and a breakdown voltage of 2 V (50% safety), this will require that d be no smaller than 10 nm.
For mica,  = 60 and  = 3 x 10**3 kg/m3 .
Hence:
W = 90 J/kg = 2.5 x10**‒2 Wh/kg.
By comparison, a lithium-ion battery has
W = 1.5 x 10**2 Wh/kg, almost 4 orders of magnitude greater
Energy density is given by:
 = permittivity of insulation material
V = applied voltage
 = density of insulation material
d = separation between plates

68. A supercapacitor is a “hybrid” battery/capacitor

69. Users of Supercapacitors

70. Energy Comparison

71. Electrostatic Potential Energy
Electrostatic potential energy density (Joules/volume)
Energy stored in a capacitor
Total electrostatic energy stored in a volume

72. Image Method
Image method simplifies calculation for E and V due to charges near conducting planes.
For each charge Q, add an image charge –Q
Remove conducting plane
Calculate field due to all charges