9.11 x 10^-31 kg
Mass of electron
1.602 x 10^-19 C
Charge of electron
Electrical, magnetic, optical, and thermal properties
For what physical properties of materials due the valence electrons largely dictate?
Drude; Maxwell-Boltzmann
___ theory treats electrons as a gas undergoing random motion and elastic collisions with no attractions, allowing for application of ___ Statistics and Distribution.
Equipartition; 3/2 kT = 1/2 mv²
The ____ Law treats the total kinetic energy of a classical particle as equal for all for 3 degrees of freedom. Meaning, the avg. translation kinetic energy of each molecule in an ideal gas is ___
v_th = (3kT/m)¹′² ; 1e5 m/s
The thermal velocity of an electron using the Drude assumption that electrons have the avg. KE of a molecular gas. At room temp, this is the approximate speed of an electron.
Positive
By convention, electric field (E) points in direction of where a ___ charge would go (+V to -V)
F = eE = ma
This is the force an electron experiences due to an electric field (E)
Drift velocity (vD = a τ); collision time
Electrons do not accelerate indefinitely due to collisions with atoms and other electrons. As a result, the electrons reach a saturation velocity called ___. The characteristic time is known as what?
e τ / m; m² / Volt-s
This collection of terms represents the electron mobility µ for the drift velocity (vD = µ E); its units are ___
Current density
The average speed and concentration of electrons, normalized per cross-sectional area
I = N e vD A
Equation for current, using concentration of electrons, charge, and drift velocity.Aka NevadA equation
Conductivity
This quantity (name only) combines electric concentration, charge, and mobility
J = σE
This version of Ohm's law uses current density
mean free path (l); No, under typical circumstances, collision time depends only on the thermal velocity (random motions)
The ___ is the distance an e- travels before subsequent collision, and is constant for a regularly spaced lattice. Does the collision time (τ) depend on vD or E?
N = val.e ⋅ mol def ⋅ ρ / Mw
What equation can we use to rough estimate the number concentration (#/vol) of electrons in a metallic element?Hint: valence e- per unit volume of materialρ in this case means density
σ = Neμ
This is conductivity in terms of charge concentration and other familiar quantities"σ new
Increases
Under typical circumstances, as temperature rises in a metal, what happens to the resistivity (ρ)?
n/N = (1/kT)^(3/2) E^(1/2) exp(-E/kT)
This equation finds the probability of finding a molecule or particle with energy E in one direction.
1.38e-23 J/K
Value of Boltzmann constant
Thermal coefficient of resistivity (TCR), α = 1/ρ₀ * dρ/dT
In this equation, what does α represent?ρ(T) = ρ₀(1+αΔT)How do you calculate it?
Matthiessen's rule
This rule states, in general, electrical resistivity is the sum of resistivities associated with conduction electron scattering by lattice vibrations (thermal vibrations), scattering due to grain boundaries/dislocations etc., and scattering by impurities. ρ = ρT + ρR + ρI
Nordheim's rule
This equation can be used to predict resistivity due to impurity. Whereρ = ρ₀ + C X (1 - X)ρi = C X (1 - X)where C = the ___ coefficient and X = mole fractionTypically a different coefficient is defined for both sides of X = 0.5
Simple addition, addition of reciprocals; volume fraction
For series mixing, how do resistivity terms add? For parallel mixing? Remember conductivity is the opposite. ___ fraction is used usually for the additions (___ᵢ ⋅ ρᵢ)
Once the dispersed phase percolates (just start touching); when the resistivity of one phase is greater than an order of mag to the other
At what point do the mixing rules for predicting resistivity (or other properties of electronic materials) no longer apply?
Remains fairly constant
Thermal conductivity ___ for metals for increasing temperature (unless really low temperatures)
F = q₀ E + q₀ v ⨯ B
This is the Lorentz relation, expressing the force a charge experiences traveling via electric and magnetic fields. Magnetic force is also commonly called Lorentz force
Positive, negative
The right hand rule is for ___ charges while the left hand rule is ___ charges. Thumb - B fieldForefinger - ForceMiddle finger - Motion
Hall Effect
This effect occurs when a magnetic field is applied normal to the E field / charge velocity in a stationary solid, and a transverse voltage results. The magnetic field causes a build of charges on one side, producing a counterbalancing E field.
E = -∇V
What is the E field in terms of voltage?∇ = (spatial) gradient
E_H = v B = J B / N q
This is the Hall electric field expressed in terms of other variables. (Hint: equate electric force with magnetic force)
Quantify N, μ, or B for metals, including semi-conductors and dielectrics
What makes the Hall effect so useful? (What useful quantities can we measure?)
A ⋅ B = ||A|| ||B|| cos θ = A₁ B₁ + A₂ B₂ + A₃ B₃
How to calculate the dot product?
A ⨯ B = ||A|| ||B|| sin θ [ i j kA₁ A₂ A₃ B₁ B₂ B₃ ]
How to calculate the cross product?What matrix do you calculate the determinant for to calculate the cross product vector?
∇ ⋅ A = ∂A₁/∂x + ∂A₂/∂y + ∂A₃/∂z
How to calculate the (spatial) divergence of a vector field?
∇ ⨯ A = <∂A₁/∂x, ∂A₂/∂y, ∂A₃/∂z>
How to calculate the curl of a vector field?
∯(∇ ⨯ A)ds = ∮AdlSurface integral, line integral
This is Stoke's Theorem using the vector field A and path line vector s. Equivalently, this theorem proposes that the ___ of the curl of a function over a closed surface is equal to the ___ of a particular vector function around that path.
The outward flux of electric field (flux) emerging from a volume must contain a net charge q equal to q/ ε₀.Gauss' Law
What does this Maxwell equation explain?Differential: ∇ ⋅ E = ρ/ε₀ Integral: ∮E ⋅ ds = q/ε₀ What is the alternative name of this law?
The sum of magnetic fields (flux) through any closed surface is 0. Gauss' Law for Magnetism
What does this Maxwell equation explain?Differential: ∇ ⋅ B = 0Integral: ∮B ⋅ ds = 0What is the alternative name of this law?
The emf around a closed curve C is equal to minus the rate of change of magnetic flux. Faraday's Law of Induction
What does this Maxwell equation explain?Differential: ∇ ⨯ E = -(∂B/∂t)Integral: ∮E ⋅ dl = -(Φ_B / dt) What is the alternative name of this law?
A magnetomotance or magnetomotive force around a closed curve C is equal to the sum of the free current plus the displacement current. Ampere's Law
What does this Maxwell equation explain?Differential: ∇ ⨯ B = μ₀ J + ε₀μ₀∂E/∂t Integral: ∮B ⋅ dl = μ₀ I + ε₀μ₀∂E/∂t What is the alternative name of this law?
B = μ H B is the magnetic induction field with units of Tesla (or Weber/m²), while H is the magnetic field with units of Ampere/meter
What is a B field versus a H field? Units for each? Equation relating the two?
4π e-7 Tesla-m/Amp
This is the value of the permeability of free space (μ₀)
D = ε EDielectric displacement has units of Coul/m² while electric field has units of Volts/m
What is the D field versus the E field? Units for each? Equation relating the two?
Flux (number of lines of force cutting through surface)
What does Φ represent in this chapter?
F = q q₀ / 4 π ε₀ r²
Derive Coulomb's Law for stationary point charges using Gauss' Law (Maxwell equation).
This means individual flux lines of B do not diverge nor converge to a single point in space, i.e., there are no magnetic monopoles.
Why is the magnetic induction field (B) sometimes said to be "divergence-free"?
Emf = Blv
What is the emf produced by the motion of this conductor (a conducting wire moving down an E field?)You could measure 1) current caused by motion of conductor in H field or 2) current caused by change in magnetic flux through stationary circuit
∇ ⨯ H = J + ∂D/∂t
What is an alternative form of Ampere's Law (in differential form)?
Same valency of the metals
What is a significant assumption of Nordheim's rule concerning the two separate metals?
F_B = q v ⨯ B = i l ⨯ B
For a DC motor, what is the magnetic force experienced by the wire?