1. ELEC 3105 Basic EM and Power Engineering Conductivity / Resistivity Current Flow Resistance Capacitance Boundary conditions

2. Conductivity and resistivity The relaxation time model for conductivity works for most metals and semiconductors. In a conductor at room temperature, electrons are in random thermal motion, with mean time  between collisions. Random motion of the electron in the metal. Electron undergoes collisions then moves off in different direction. electron collision

3. Conductivity and resistivity The relaxation time model for conductivity works for most metals and semiconductors. In a conductor at room temperature, electrons are in random thermal motion, with mean time  between collisions. Electrons acquire a small systematic velocity v* component in response to applied electric field electron collision

4. Conductivity and resistivity For a weak electric field v* can be obtained. m = mass of electron  = carrier mobility (ELEC 2507) electron collision {  } units of

5. Conductivity and resistivity FOR A STRONG ELECTRIC FIELD for low fields v* proportional to E For strong electric fields, electrons acquire so much energy between collision that mean time between collisions is reduced. v* E

6. Conductivity and resistivity As long as we stay in the weak electric field regime, i.e. the linear region of the curve in the previous slide, then the current density can be defined as: v* E This region Conductivity Resistivity

7. Conductivity of elements

10. Current flow The total amount of charge moving through a given cross section per unit time is the current, usually denoted by I. Conductor ??? vdt dq v CURRENT

11. Current flow If we consider the current per unit cross-sectional area, we get a value which can be defined any point in space as a vector, typically denoted vdt dq v cross-sectional area A N charged particles per unit volume moving at v meters per second dq = N q vdt A Charge moving through cross-sectional area A in time dt

12. Current flow The charge density is simply this quantity divided by the unit time and area. The current density is: vdt dq v cross-sectional area A N charged particles per unit volume moving at v meters per second dq = N q vdt A Charge moving through cross-sectional area A in time dt dq = N q vdt A

13. Current flow The total current through the end face can be obtained from the current density as an integration over the cross-sectional area of the conducting medium. vdt dqv cross-sectional area A TOTAL CURRENT

14. Current flow The total charge passing through the cross-sectional area A over a time interval from t 1 to t 2 can be obtained from: vdt Q v cross-sectional area A TOTAL CHARGE

15. MOSFET

17. Resistance of conductors: any shape RESISTANCE

18. Resistance of conductors: any shape A uniform rectangular bar Electric field is uniform and in the direction of a bar length L.

19. Resistance of conductors: any shape A uniform rectangular bar Electric field is normal to the cross- sectional area A.

20. Resistance of conductors: any shape A uniform rectangular bar

22. SUPERCONDUCTORS

23. Capacitance Capacitance is a property of a geometric configuration, usually two conducting objects separated by an insulating medium. Capacitance is a measure of how much charge a particular configuration is able to retain when a battery of V volts is connected and then removed. The amount of charge Q deposited on each conductor will be proportional to the voltage V of the battery and some constant C, called the capacitance. Capacitance {C/V}

24. Parallel plate capacitor Free space between plates +Q -Q V = 0 volts V = V volts z Plate area A Plate separation D Between plates At z = D Rearrange Capacitance of parallel plate capacitor

25. CAPACITORS IN SERIES/ PARALLEL/ DECOMPOSITION PARALLEL C1C1 C2C2 C eq C1C1 C2C2 SERIES C eq

26. CAPACITORS IN SERIES/ PARALLEL/ DECOMPOSITION DECOMPOSITION C eq C1C1 C2C2 C3C3

27. CAPACITANCE OF A COAXIAL TRANSMISSION LINE Prove this result as part of next assignment. If we consider as the charge per unit length on each of the two coaxial surface, then: (ELEC 3909)

30. CHARGE CONSERVATION AND THE CONTINUITY EQUATION Charge in volume v Current through surface A Also recall The main ingredients to the pie

31. 31 CHARGE CONSERVATION AND THE CONTINUITY EQUATION Then: Current out of volume is From divergence theorem Using previous expressions

32. CHARGE CONSERVATION AND THE CONTINUITY EQUATION Interpretation of equation: The amount of current diverging from am infinitesimal volume element is equal to the time rate of change decrease of charge contained in the volume. I.e. conservation of charge. In circuits: If no accumulation of charge at node.

33. 33 CHARGE CONSERVATION AND THE CONTINUITY EQUATION A charge is deposited in a medium. Also

34. CHARGE CONSERVATION AND THE CONTINUITY EQUATION A charge is deposited in a medium. If you place a charge in a volume v, the charge will redistribute itself in the medium (repulsion???). The rearrangement of charge is governed by the constant T = REARRANGEMENT TIME CONSTANT T Cu, Ag =10 -19 sT mica =10 h

35. Boundary conditions Tangential Component of Around closed path (a, b, c, d, a) ELECTROSTATICS Boundary a b c d Potential around closed path

36. Boundary conditions Tangential Component of Boundary a b c d ELECTROSTATICS

37. Boundary conditions Tangential Component of a b c d The tangential components of the electric field across a boundary separating two media are continuous. ELECTROSTATICS

38. Boundary conditions Tangential Component of At the surface of a metal the electric field can have only a normal component since the tangential component is zero through the boundary condition. ELECTROSTATICS a b c d metal

39. Boundary conditions Normal Component of Gauss’s law over pill box surface ELECTROSTATICS Boundary

40. ELECTROSTATICS Boundary Boundary conditions Normal Component of

41. 41 The normal components of the electric flux density are discontinuous by the surface charge density. Boundary conditions Normal Component of ELECTROSTATICS

42. 42 Boundary conditions Normal Component of ELECTROSTATICS metal At the surface of a metal the electric field magnitude is given by E n1 and is directly related to the surface charge density.

43. Boundary conditions Normal Component of ELECTROSTATICS Gaussian Surface Air Dielectric Gaussian surface on metal interface encloses a real net charge  s. Gaussian surface on dielectric interface encloses a bound surface charge  sp, but also encloses the other half of the dipole as well. As a result Gaussian surface encloses no net surface charge.

44. ELEC 3105 Basic EM and Power Engineering Extra extra read all about it! 44

45. Electric fields in metals 45

46. Electric fields in metals (a) no current E inside = 0 (b) with current E inside  0

47. Inhomogeneous dielectrics We can consider an inhomogeneous dielectric as being made up of small homogeneous pieces, at the interfaces of which bound charge will accumulate. x Suppose that we have a dielectric whose permittivity is a function of x, and a constant D field is directed along x as well. dielectric

48. Inhomogeneous dielectrics We can consider an inhomogeneous dielectric as being made up of a stack of thin sheets of thickness dx and permittivity  (x). x In each sheet, positive charges will accumulate on the right and negative ones on the left, according to the permittivity of the sheet.

49. Inhomogeneous dielectrics We can consider an inhomogeneous dielectric as being made up of a stack of thin sheets of thickness dx and permittivity  (x). x The charges will mostly cancel by adjacent sheets, but any difference in permittivity between adjacent sheets d  will leave some net charge density.

50. Inhomogeneous dielectrics We can consider an inhomogeneous dielectric as being made up of a stack of thin sheets of thickness dx and permittivity  (x). x We can express this net bound charge easily as the difference in polarizations, so that we have:

51. Inhomogeneous dielectrics In the more general case when the permittivity is varies in all directions, i. e.  (x,y,x). x We can express this net bound charge easily as the difference in polarizations, so that we have: y z

52. Inhomogeneous dielectrics In the more general case when the permittivity is varies in all directions, i. e.  (x,y,x). x Take divergence on each side: y z