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

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. electron collision Random motion of the electron in the metal. Electron undergoes collisions then moves off in different direction.

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. electron collision Electrons acquire a small systematic velocity v* component in response to applied electric field

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

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

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

Conductivity of elements

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

Current flow dq = N q vdt A 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 cross-sectional area A dq v vdt 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

Current flow dq = N q vdt A dq = N q vdt A The charge density is simply this quantity divided by the unit time and area. The current density is: dq = N q vdt A cross-sectional area A dq v vdt 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

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. cross-sectional area A dq v vdt TOTAL CURRENT

Current flow The total charge passing through the cross-sectional area A over a time interval from t1 to t2 can be obtained from: cross-sectional area A Q v vdt TOTAL CHARGE

MOSFET

Resistance of conductors: any shape

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

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

Resistance of conductors: any shape A uniform rectangular bar

SUPERCONDUCTORS

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}

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

CAPACITORS IN SERIES/ PARALLEL/ DECOMPOSITION Ceq Ceq

CAPACITORS IN SERIES/ PARALLEL/ DECOMPOSITION Ceq C1 C2 C3

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)

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

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

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.

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

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 TCu, Ag=10-19 s Tmica=10 h

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

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

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

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

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

Boundary conditions Normal Component of ELECTROSTATICS Boundary conditions Normal Component of Boundary

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

Boundary conditions Normal Component of ELECTROSTATICS Boundary conditions Normal Component of metal At the surface of a metal the electric field magnitude is given by En1 and is directly related to the surface charge density.

Boundary conditions Normal Component of ELECTROSTATICS Boundary conditions Normal Component of 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. Air Dielectric Gaussian Surface

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

Electric fields in metals

Electric fields in metals (a) no current Einside = 0 (b) with current Einside  0

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. Suppose that we have a dielectric whose permittivity is a function of x, and a constant D field is directed along x as well. x dielectric

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

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

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

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

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