of Fundamentals Noise Processes
EE248/Fall 2007 Instructor: Professor Yoshihisa Yamamoto
Please address homework questions to Kristiaan De Greve ([e-mail]).
Problem Set 3
Due date: November 5, 2008
Problem I: External Circuit Current (Rs = 0 case) (15 points) Look at the vacuum diode (Fig.1). Suppose an electron is emitted from the cathode and is in transit to the anode. Assume the source resistance Rs is zero. Show that the external short-circuit current due to this moving charge carrier is given by i(t) = qv(t) d
where q is the fundamental unit of charge, v(t) is the electron drift velocity, and d is the distance between the two electrodes. HINT: Calculate the energy gained by the electron from the electric field associated with this moving charge, and equate it to the energy provided by the external constant applied voltage V. Problem II: Surface Charges and External Circuit Current (25 points) In a vacuum diode (Fig.1), a single electron is emitted from the cathode at t = 0. For each of the following three cases, calculate the surface charges of the cathode and anode, as well as the external circuit current as a function of time. (1) The electron drift velocity is initially zero at the cathode and is accelerated by the constant applied electric field, and RC t . This might be the case if the initial kinetic energy of the electron is small relative to the energy gained in transit from the field. (2) The electron drift velocity is assumed to be constant over the electron's transit from the cathode to the anode, as is the case if the initial kinetic energy is large relative to qV . In addition, assume RC t . (3) RC t . In this case, we assume the electron transit to be an impulsive event.
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2
cathode charge QC(t) -q v anode charge QA(t) i(t)
capacitance C d
Rs V
Figure 1. A vacuum diode. Problem III: Equipartition theorem (30 points) A small system A, with allowed energies 1 , 2 , ..., is placed in thermal contact with a heat reservoir B. In other words, although their total energy E0 remains constant, A and B are free to exchange energy. It can be shown that the probability distribution for finding system A with energy i is P ( i) = e- i /kB - i /kB ie
Here, kB is Boltzmann's constant, and is the temperature of the reservoir B. This distribution is known as the canonical distribution, and characterizes the distribution for a system that can exchange energy, but not particles, with a reservoir. Note that the probability of occupation decreases exponentially with the energy of the eigenstate; the exponential in the numerator is known as the Boltzmann factor. (1) Show that the probability of finding the system A with energy i is given by the canonical distribution above. HINT: The key to this probelm is to note that if system A has energy i , system B must have energy E0 - i . We postulate that the probability of system A having energy i is proportional to the number of internal microstates of the reservoir with energy E0 - i , which we label as (Eo - i ). Now, as system A is much smaller than B, we assume that i E0 , implying that log[(Eo - i )] can be expanded as a power series around E0 . Note also that two systems with system energies Ei , Ej are considered to be in thermal equilibrium (and hence have the same temperature ) if and only if the derivatives of ln i (Ei ) and ln j (Ej ) with respect to their respective system energies Ei , Ej are equal. From that, an appropriate definition of the temperature can be derived, which you will need in your derivation. (2) Let us now say that the system of system can A be described by a set of continuous, classical dynamical variables x1 , x2 , ..., xn and corresponding momenta p1 , p2 , ..., pn . (As an example, A might consist of a collection of n/3 gaseous
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molecules, and {xi , pi } may represent the position and momentum of each molecule along a prescribed coordinate axis.) We make two assertions: The energy of system A can be split into two terms, only one of which is dependent on pi : = E(pl ) + (q1 , q2 , ..., qn ; p1 , p2 , ..., pl-1 , pl+1 , ..., pn ) E(pl ) is quadratic in pl : E(pl ) = bp2 l where b is a given constant. Under these conditions, show that the statistical average of E(pl ) is given by This result is known as the equipartition theorem. It states that the mean 1 value of each independent quadratic term in the energy is 2 kB . HINT: The statistical average of quantity a is given by
1 k . 2 B
a =
aP (a)dq1 ...dqn dp1 ...dpn .
(For this formula to be appropriate, we must assume that the density of accessible states in phase space is constant.) (3) If A is a quantum mechanical system with a discrete eigenenergy spectrum, does the equipartition theorem hold ? Discuss the conditions under which your result is valid. Problem IV: Macroscopic conductor (30 points) At finite temperature, the electron collisions with the lattice vibrations and impurities induce fluctuations in the electron's velocity and position. The electronic motion is well explained by a Brownian particle model. The net microscopic electronic thermal agitation drives a fluctuating voltage across the resistor. In this problem, you will derive the Johnson-Nyquist formula for equilibrium thermal noise using the fluctuationdissipation theorem. Consider a parallel resistor-inductor circuit. R is the resistance and L is the inductance. Given the voltage across the resistor V0 (t), the Kirchhoff voltage law tells you that L dI(t) = V0 (t), where I(t) is the current of the circuit. Suppose V0 (t) decomposes dt into a slowly-varying part V (t) and a rapidly-varying part v(t). (1) Show that the Langevin equation for this circuit is L dI(t) = -RI(t) + v(t) dt
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I (t)
+
V0 (t)
-
R
L
Figur...