Computer Lab 4: Lecture 11

In our computer lab lectures we will work together through concepts and challenges we have discussed over the week in class.  As with your after-class Exercises, you will be expected to turn in your computer lab notebooks before the nect lecture. Please do so by e-mailing your completed *nb file to Lan Yu (ece329.uiuc@gmail.com), make sure you name the file using the following notation: “Exn_Last_First.nb”, where n is the Lecture number (in this case, 11), and First and Last would be your first and last name.  So my submission to Lan for this computer lab would be titled “Ex11_Wasserman_Dan.nb”.  You will receive full credit for your copmputer lab if it turned in before the next lecture (Lecture n+1).  No late submissions will be accepted.

Objective/Overview:

In class this week we discussed the behavior of materials (conductors and dielectrics) in the presence of electrostatic fields.  We then used this knowledge to start looking at our first real electrical devices (capcitors).  We used our ability to relate Electric Field, Charge, and Electrostatic Potential to understand the relationship between charge and potential (which gives us capcitance).  We also were able to describe the conductance associated with our capcitive structures.  We looked at both parallel plate capacitors, as well as coax structures, and were able to determine capcitance and conductance (both total, and per unit length) of these structures.
Up to know we have considered only static fields (thus electrostatics).  However, as we move forward in the course, we will begin working more and more with time-dependent fields.  Today we will take our first look at time dependent fields, and in particular, how these fields interact with matter!

ClearAll[sol,normsigma,ef,j, normj, x, normx, x1]

Motion of Charges in an Electric Field

In our last lecture, we discussed how small changes in charge can lead to changes in electrostatic potential (voltage) in our electrostatic systems.  Of course, these changes occur over some time frame, so this was our first look at what happens when we start changing charge/fields/voltages in our electrostatic systems.  
Let’s start our discussion of the interaction of charges and time-dependent field by first looking at how a charge behaves in the presence of a uniform electric field.  We can start by writing out Newton’s Force Law
(OverVector[F])=m(OverVector[a])=q(OverVector[E])-((m(OverVector[v]))/τ)
This expression, of course, relates the acceleration of a charge q as a function of the forces exerted on the charge. You will recognize the m(OverVector[a]) term from Intro Physics and the q(OverVector[E]) as the force exerted on the charge from the electric field.  The last term, ((m(OverVector[v]))/τ), can be thought of as a “damping” term, kind of like air resistance for a falling object.  In the case of electrons, we can think of this term as coming from scattering of the electron (off of other electrons, defects in the material, or from lattice vibrations).  The τ term is the average time an electron travels before scattering.  Once the electron scatters, it’s velocity renormalizes...namely, the elastic scattering process conserves energy, but not momentum, so the electron can scatter in ANY direction. Thus, the average velocity vector after scattering is 0.  So every τ seconds, the electron loses m(OverVector[v]) of momentum.  This change in momentum is an “impulse”, and the average “impulse” over time is nothing more than the “damping” Force exerted on the electron.
So we now have a rather simple differential equation that we should be able to solve for a variety of conditions.  Let’s start by looking at the case where we have some initial velocity “vo”, and no electric field.

sol=DSolve[{m*v'[t]==-m*v[t]/tau,v[0]==vo},v[t],t]

Cool!  We see that the velocity of our charge decays exponentially from its initial value of vo, with a characteristic decay time τ.
What about if we have a constant electric field?  In this case our charge will accelerate, and as it gains speed, the damping term will increase.  Eventually the force from the electric field will be entirely balanced by the damping term.  This will make our acceleration 0, and is what we call “steady state”: it is not an equilibrium situation, but the average motion of the charge will not change over time...

sol=DSolve[{0==q*efield-m*v[t]/tau},v[t],t]

Aha!  We see that our velocity is now constant, and given by: (OverVector[v])=((q(OverVector[E])τ)/m)
Our velocity is proportional to the electric field!  That constant of proportionality is known as the “mobility”, μ=((qτ)/m)
If we assume that instead of one charge, we have a density of charges: “N”, in units of (m^(-3)), then we can write the expression for current density: (OverVector[J])=Nq(OverVector[v])=Nq((qτ)/m)(OverVector[E])=σ(OverVector[E]).
We can see that this parameter, conductivity, which we introduced earlier in the week, can actually be derived simply from Newton’s Force Law!

Motion of Free Charges in a Time-Varying Electric Field

Now let’s see what happens when we introduce a time varying field (OverVector[E])(t)=Re[(OverTilde[E])(e^(jωt))].  Here we are using “phasor” notation, which we will use a lot in the coming months. Using phasor notation, we recognize (OverTilde[E]) as the complex amplitude of our electric field, and (e^(jωt)) as the harmonic time-dependence of the field.  Remember from Euler’s equation, that (e^(jωt))=cosωt+jsinωt.  We can also write our current density as (OverVector[J])(t)=Re[(OverTilde[J])(e^(jωt))].  One of the reasons we use phasor notation is that taking the time derivative of a phasor is remarkably simple: (∂/(∂t))Re[(OverTilde[v])(e^(jωt))]=Re[jω(OverTilde[v])(e^(jωt))].
If we assume that the velocity of the charge has the same time dependence as the driving electric field (why wouldn’t it?), we can also write (OverVector[v])(t)=Re[(OverTilde[v])(e^(jωt))].  We can now plug this all into Newton’s Force Law:
(OverVector[F])=m(OverVector[a])=q(OverVector[E])-((m(OverVector[v]))/τ)
m(∂/(∂t))Re[(OverTilde[v])(e^(jωt))]=mRe[jω(OverTilde[v])(e^(jωt))]=qRe[(OverTilde[E])(e^(jωt))]-(m/τ)Re[(OverTilde[v])(e^(jωt))]
We can drop the “Re” here to give us the complex velocity and electric field, and then divide by (e^(jωt)) on both sides to relate the complex phasors (OverTilde[v]) and (OverTilde[E]).
mjω(OverTilde[v])=q(OverTilde[E])-(m/τ)(OverTilde[v])
We can now relate the velocity of the charge to the electric field
(OverTilde[v])=(q/(m(jω+1/τ)))(OverTilde[E]), (OverTilde[J])=Nq(OverTilde[v])=((Nq^2)/(m(jω+1/τ)))(OverTilde[E]), σ=((Nq^2)/(m(jω+1/τ)))
We see that for time-dependent fields, we have a complex(!) conductivity!  Here, τ is our scattering time, and γ=1/τ we call a scattering frequency.  Let’s plot our conductivity as a function of frequency ω, so that we can understand the effect of scattering time and frequency on the response of our system of charges.  We will “normalize” sigma so that we don’t have to worry about charge and charge density.  In otherwords, we will plot sigma/(N(q^2)/m).

normsigma[gamma_]:=1/(I*w+gamma)

Manipulate[Plot[{Re[normsigma[gamma]],Im[normsigma[gamma]], Abs[normsigma[gamma]]},{w,0,1000},PlotLegends->{"Real","Imag.","Abs[σ]"},AxesLabel->{"frequency","σ"}],{gamma,{0,w/100,w/10,w,10w}}]

For gamma=0 (infinite scattering time, or essentially no scattering), we see that the complex conductivity of our material is entirely imaginary, and decreases in magnitude with increasing frequency.  If our gamma is larger than omega, then we see that our conductivity is primarily real and positive.  What does this mean in terms of the motion of the charge?  We can figure this out!  Let’s assume that N(q^2)/m=1, and that the amplitude of our incident electric field is real, (OverTilde[E])=1.  Now let’s plot current as a function of time for different frequencies, for a fixed gamma=10.

ef[t_,w_]:=Re[Exp[I*w*t]]

j[t_,w_]:=Re[(1/(I*w+10))*Exp[I*w*t]]

Manipulate[Plot[{ef[t,w],j[t,w]},{t,0,If[w==0,1,10/w]},PlotLegends->{"Electric Field","Current"},AxesLabel->{"time","Ef or J"}],{w,{0,.1,1,10,100}}]

This plot shows the relationship between current and electric field (both phase and amplitude).  But let’s focus on phase by normalizing the current...

normj[t_,w_]:=Re[(1/(I*w+10))*(Abs[I*w+10])*Exp[I*w*t]]

Manipulate[Plot[{ef[t,w],normj[t,w]},{t,0,If[w==0,1,10/w]},PlotLegends->{"Electric Field","Normalized Current"}],{w,{0,.1,1,10,100}}]

Really quickly, let’s also plot the displacement of the charges, which we know will be related to the integral of the velocity

x[t_,w_]:=Re[(1/(I*w+10))*Exp[I*w*t]*(1/(w*I))]

normx[t_,w_]:=Re[(1/(I*w+10))*Exp[I*w*t]*(1/(w*I))*(Abs[(I*w+10)*I*w])]

Manipulate[Plot[{ef[t,w],normx[t,w]},{t,0,10/w},PlotLegends->{"Electric Field","Displacement"}],{w,{0.01,.1,1,10,100}}]

What does this tell us?
Well, when ω>>γ, namely when our electric field is changing much faster than the charges can scatter, we see that our displacement is out of phase with the electric field, and our velocity is actually 0 when the field is greatest!  
When ω<<γ, namely our electric field is changing much slower than the charges can scatter, our velocity tracks with the field, as would be expected from what we know about DC conductivity.  In this situation, you can think of the electric field as changing slowly, such that the electrons can reach their ‘terminal velocity’, which will scale with electric field.  Thus, the electron velocity will follow the field.

Motion of Bound Charges in a Time-Varying Electric Field

Now let’s see what happens when we introduce a time varying field (OverVector[E])(t)=Re[(OverTilde[E])(e^(jωt))] to a system of bound charges, which is what we have in a dielectric.  In this case, σ=0, but we have some susceptibility that leads to a relative permittivity (Subscript[ϵ, r])=1+(Subscript[χ, e]).  In this case, let’s go back to basics, and assume our bound charge is tied to the nucleus by a spring, with spring constant k.  We know, from intro physics, that a spring mass system has a resonance frequency (Subscript[ω, o])=(Sqrt[k/m]), so k=(Subscript[mω, o]^2)  Keeping a damping term proportional to velocity, which we will describe with a scattering time for the dielectric (Subscript[τ, d]), we can write Newton’s Law now as: 
(OverVector[F])=m(OverVector[a])=-e(OverVector[E])-((m(OverVector[v]))/(Subscript[τ, d]))-(Subscript[mω, o]^2)x
m(OverVector[x])''=-e(OverVector[E])-((m(OverVector[x])')/(Subscript[τ, d]))-(Subscript[mω, o]^2)x
For a constant electric field, we can assume no acceleration and no velocity (bound charge in constant field) and solve for x, our displacement:

sol=DSolve[{-e*ef==m*wo^2*x1[t]},x1[t],t]

This is just the displacement of the electron from the nucleus under a constant electric field, which is what we used to calculate permittivity of dielectrics.  Using this, we can find the susceptibility of our dielectric material using the microscopic properties related to scattering and “spring constant”:
(OverVector[p]) = -e (OverVector[r])=((e^2)/(Subscript[mω, o]^2))(OverVector[E])
(OverVector[P])=(Subscript[N, d])(OverVector[p]) =(((Subscript[N, d])(e^2))/(Subscript[mω, o]^2))(OverVector[E])=(Subscript[ϵ, o])(Subscript[χ, e])(OverVector[E]), so
(Subscript[χ, e])=(((Subscript[N, d])(e^2))/((Subscript[mϵ, o])(Subscript[ω, o]^2))), which is our DC susceptibility!
What about the AC susceptibility?  Well, we can go back to our phasor notation to describe the time dependence of our position, velocity, acceleration, and electric field.
m((∂^2)/(∂(t^2)))Re[(OverTilde[x])(e^(jωt))]=mRe[-(ω^2)(OverTilde[x])(e^(jωt))]=-eRe[(OverTilde[E])(e^(jωt))]-(m/τ)Re[jω(OverTilde[x])(e^(jωt))]-(Subscript[mω, o]^2)x, rearranging in the same way we did for the DC case gives:
(OverTilde[x])=((-e/m)/(((Subscript[ω, o]^2)-(ω^2))+jωγ))(OverTilde[E]), and (Subscript[χ, e])=(((Subscript[N, d])(e^2)/(Subscript[ϵ, o])m)/(((Subscript[ω, o]^2)-(ω^2))+jωγ))
You can see that as ω-->0 (DC), our susceptibility goes to the DC solution we derived above.  What does our bound electron displacement look like as a function of the frequency of the electric field?  That’s easy to compute.  Let’s simplify things and set e/m=1, (Subscript[ω, o])=10 and assume the amplitude of our incident electric field is real, (OverTilde[E])=1 :

x1[w_,gamma_]:=-1/((10^2-w^2)+I*w*gamma)

Manipulate[Plot[{Re[x1[w,gamma]],Im[x1[w,gamma]]},{w,0,20},PlotLegends->{"Real{x}","Imag.{x}"},AxesLabel->{"w","x1"},PlotRange->Full],{gamma,{0.01,.1,1,10,100}}]

This, of course, is nothing more thanthe frequency response of a damped harmonic oscillator.  In such a system, we can think of the real part of x as the amplitude of the displacement from equilibrium, and the imaginary part of x is actually related to energy loss, which we can see is resonant at (Subscript[ω, o]).  Intuitively, it makes sense that the displacement of the electron is opposite in sign to the electric field, since the force on the electron from the E field is F=-eE.

Polarization Current

The world is very different when we allow the electric field to change in time. In particular, for dielectrics, we know that for an electrostatic field no current can flow, since there are no free carriers to carry current.  However, for time-varying electric fields, this is not the case.  For a time varying field, we clearly see that we are ‘sloshing’ around our bound charges...they oscillate at the same frequency as the incident electric field.  Using this, and the fact that (OverVector[J])=-(Subscript[eN, d])(OverVector[v]), use our above derivation for (OverTilde[x])=((-e/m)/(((Subscript[ω, o]^2)-(ω^2))+jωγ))(OverTilde[E]) to find (OverVector[x])(t) and determine the (AC) current from bound charge in our dielectric.  This current is the Polarization Current associated with the motion of bound charge, and will play an important role in our final Maxwell’s Equations. Assume  N(e^2)/m=1, (Subscript[ω, o])=10, and (OverTilde[E])=1.  Design a “Manipulate” plot showing E(t) and Jpol(t), where you can change the gamma, and the frequency.

jpol[t_,w_,gamma_]:=

ef[t_,w_]:=Re[Exp[I*w*t]]

Manipulate[Plot[{jpol[t,w,gamma],ef[t,w]},{t,0,4},PlotLegends->{"Polarization Current","Electric Field"},PlotRange->Full,AxesLabel->{"t","E or Jpol"}],{gamma,{0.01,.1,1,10,100}},{w,0,20}]

Use this plot to describe below (briefly) the phase and amplitude of the polarization current for the following cases:
1) ω=(Subscript[ω, o]), little to no scattering (weakly damped)
2) ω>>(Subscript[ω, o]), weakly to moderately damped γ≤(Subscript[ω, o])