ELECTRICAL FLUCTUATIONS AND THE SENSITIVITY LIMIT OF ELECTRICAL INSTRUMENTS

V. L. Granovskii, Moscow

I. Thermal Effect

Theoretical Concepts

1. The conception of the atomic structure of electricity (Helmholtz), which arose in the last quarter of the previous century, subsequently received its complete and comprehensive confirmation in the development of experimental physics. The study of the carriers of electricity in solid, liquid, and gaseous media, both conducting and dielectric, showed that in all these cases electricity appears in the form of individual atoms, which were given the name electrons*. The very same electrons were discovered inside the atoms of all chemical elements as their constituent parts. The magnitude of the electron, determined by various methods, always proved to be one and the same and equal (according to Millikan) to \(e = 1.591 \cdot 10^{-19}\) coulomb. At present there are apparently no grounds for assuming the existence of smaller electric charges.

The establishment of the atomicity of electricity entailed the introduction of kinetic concepts into the theory of electricity. The founders of the classical electron theory—J. Thomson, Drude, and Lorentz—constructed the theory of electrical phenomena in metals on the basis of the idea that atoms of electricity participate in thermal motion alongside the atoms of the metal. Electrons in a conductor never remain at rest; on the contrary, they move with various and randomly directed velocities in all parts of the metal. Such motion does not disappear even in the presence of a total current; the imposition of an electric field causes an additional general displacement of all the electrons, which is superposed on their disordered thermal motion. This

* This term, which originally denoted simply an elementary quantity of electricity, irrespective of its sign and of the mass associated with it, was subsequently specialized to designate only atoms of negative electricity. At the present time (1933) a return to the original use of this word is being observed; in what follows we shall adhere precisely to this meaning of the word “electron.”

picture made it possible to construct a theory of the metallic electrical and thermal conductivities that satisfactorily explains a number of the basic regularities in this field. The construction of the electronic theory of metals is based on the same methods on which the kinetic theory of matter is built—on the methods of statistics. The chaotic motion of the numerous electrons is subject to the laws of chance; the study of these motions can be carried out only with the aid of statistics based on the theory of probability. With this approach, all quantities characterizing the behavior of electrons in bulk, such as: current, charge density, potential difference in any part of a conductor, turn out to be statistical quantities with all the peculiarities of the latter.

1. One of the most characteristic features inherent in all statistical quantities is spatial-temporal nonconstancy. As a simple example, let us take the number of molecules occurring in \(1\ \mathrm{mm}^3\) of gas under normal conditions. This number is on the average equal to

\[ \nu=\frac{6.06\cdot 10^{23}}{22400\cdot 10^{3}}=2.71\cdot 10^{16}. \]

However, this by no means signifies that in every cubic millimeter of some large vessel there are exactly that many molecules. On the contrary, the theory of probability teaches that such a uniform distribution of molecules over separate regions is extremely improbable. If at some moment in time we could make an instantaneous photograph of the volume of gas under study, and one such that all the molecules were visible on it, then we would find the following distribution of them among the individual cells. Most frequently there would occur cells in which there were \(\nu=2.71\cdot 10^{16}\) molecules; however, there would also be cells in which the number of molecules \(n\) is less or greater than \(\nu\). Moreover, the more strongly \(n\) differs from \(\nu\), the fewer cells with the corresponding number of molecules we would find. The same result could be obtained in another way: namely, instead of determining once the distribution of molecules throughout the whole volume, we could determine the number of molecules only in one cell, but repeat this very many times. We would find that most often this cell contains precisely \(2.71\cdot 10^{16}\) molecules; it happens, however, that extra molecules fall into it, or, conversely, that some number of molecules is lacking up to the average. The frequency of observing \(n\) molecules in a given volume is a function with a maximum at \(n=\nu\), falling sharply on both sides of the maximum and rapidly tending to zero. If this whole picture is translated into the language of macroscopic quantities, then the number of molecules per unit volume will represent the density of the gas. We shall then formulate the preceding result as follows: the density of the gas at different places undergoes small, irregular fluctuations around the mean value. The cause of these fluctuations

is the thermal motion of molecules; in the absence of motion the density would remain everywhere unchanged. Such oscillations affect not only the density of the medium; other parameters, for example, temperature, pressure, etc., are also statistical quantities, and likewise are fully determined only on the average. These oscillations of macroscopic parameters, occurring as a consequence of molecular motions, bear the general name of fluctuations. The theory of fluctuations was developed by Einstein and Smoluchowski¹; it proved extraordinarily fruitful, since it made it possible to explain the origin and regularities of a number of phenomena—the blue color of the sky, the scattering of light by substances in the critical state (opalescence), Brownian motion. The formulae derived from this theory made it possible, on the basis of various experimental data, to determine the number of molecules in one gram-molecule \(N\)—this fundamental constant of atomic physics.

The laws of statistical physics lead to mathematical expressions determining the probability of fluctuations. Let us suppose that we are interested in some parameter pertaining to a given mass of gas. Let its mean value be \(\lambda_0\). What is the probability that it will assume a value \(\lambda \ne \lambda_0\)? The answer to this question can be obtained from Boltzmann’s formula, which relates the probability of a given state of a gas to its entropy. If the probability of the state in which the parameter under consideration has a value equal to \(\lambda\) is denoted by \(W(\lambda)\), and the entropy of this state by \(S\), then

\[ S = k \ln W, \tag{1} \]

where \(k=\dfrac{R}{N}\) is Boltzmann’s constant, equal to \(1.37 \cdot 10^{-16}\) erg·degree.

In order to transfer the gas from this state to the normal state, in which the parameter is equal to \(\lambda_0\), it is in general necessary to perform a certain work \(A\); if the process is to take place at constant energy, then the corresponding amount of heat \(Q\) equivalent to \(A\) must be removed from the gas. From thermodynamics it is known that

\[ A = \int dE - \int T\,dS, \]

where \(E\) is the energy, \(T\) the absolute temperature. But \(dE=0\) by assumption; further, since in practice one has to deal only with the largest deviations from mean quantities, the temperature \(T\) may be considered constant during this small transition. Then

\[ A = -T(S-S_0), \tag{2} \]

which represents the magnitude of the change in the free energy of the gas.

From formulas (1) we find:

\[ S-S_0=k\ln\frac{W}{W_0}; \]

substituting this expression into (2), we obtain:

\[ A=-kT\ln\frac{W}{W_0}, \]

whence

\[ W=W_0 e^{-\frac{A}{kT}} . \tag{3} \]

The work \(A\), obviously, is a function of the parameter \(\lambda\); the more strongly \(\lambda\) differs from \(\lambda_0\), the greater the work required for the transition between these states. Therefore \(A\) may be expanded in a Maclaurin series in powers of \((\lambda-\lambda_0)\). Since \(\lambda=\lambda_0\) corresponds to the state of equilibrium, this series begins only with the term of the 2nd order:

\[ A=a(\lambda-\lambda_0)^2+\cdots \]

Fig. 1.

As was already noted above, we restrict ourselves only to small deviations. This can now be justified a posteriori on the basis of formula (3). It is clear from it that for large deviations, when \(A\) is large, the probability \(W\) is extremely small (\(A\) enters the exponent). Therefore, if we treat such rarely occurring cases inexactly, the general rigor of the theory will suffer little. Making use of this, we restrict ourselves in the expansion of \(A\) to the first term and substitute it into formula (3); we find:

\[ W=W_0 e^{-\frac{a}{kT}(\lambda-\lambda_0)^2}. \tag{3'} \]

Strictly speaking, it is incorrect to pose the question of the probability that the parameter has a value exactly equal to \(\lambda\). It is more rational to seek the probability that the parameter under consideration lies within the limits from \(\lambda\) to \(\lambda+d\lambda\). The probability of this is

\[ W(\lambda)\,d\lambda = W_0 e^{-\frac{a}{kT}(\lambda-\lambda_0)^2}\,d\lambda . \tag{3'} \]

The law obtained is analogous to Gauss’s law of distribution of random errors; its graphical representation has the form shown in Fig. 1.

To determine the magnitude \(W_0\), we shall use the circumstance that the sum of the probabilities of all deviations is equal to unity:

\[ \int_{-\infty}^{+\infty} W(\lambda)\,d\lambda = \int_{-\infty}^{+\infty} W_0 e^{-\frac{a}{kT}(\lambda-\lambda_0)^2}\,d\lambda = W_0\sqrt{\frac{\pi kT}{a}} = 1. \]

Hence

\[ W_0=\sqrt{\frac{a}{\pi kT}}. \tag{3''*} \]

Let us now calculate what the mean value of \(A\) is, i.e., how much, on the average, the free energy of the system differs from the normal value. We shall denote mean quantities by horizontal bars over the letters:

\[ \overline{A}=a\,\overline{(\lambda-\lambda_0)^2} =a\int_{-\infty}^{+\infty}(\lambda-\lambda_0)^2 W(\lambda)\,d\lambda = \]

\[ =\int_{-\infty}^{+\infty} a(\lambda-\lambda_0)^2 W_0 e^{-\frac{a}{kT}(\lambda-\lambda_0)^2}\,d\lambda. \]

Substituting for \(W_0\) its value found above, we can transform the integral to the form:

\[ \frac{kT}{\sqrt{\pi}}\int_{-\infty}^{+\infty} \frac{a}{kT}(\lambda-\lambda_0)^2 e^{-\frac{a}{kT}(\lambda-\lambda_0)^2} \,d\left[\sqrt{\frac{a}{kT}}(\lambda-\lambda_0)\right] = \]

\[ =\frac{kT}{\sqrt{\pi}}\int_{-\infty}^{+\infty}\xi^2 e^{-\xi^2}\,d\xi. \]

The last integral is equal to \(\frac{1}{2}\sqrt{\pi}\), consequently,

\[ \overline{A}=\frac{kT}{2}. \tag{4} \]

This remarkable formula shows that for any gas under any conditions the mean energy of fluctuations is equal to the mean energy of the translational motion of one molecule in a definite direction. The result obtained may be regarded as a certain extension of the law of the uniform distribution of energy over degrees of freedom; not only the energy of each degree of freedom of one molecule, but also the energy of each

* The integral written in the text may be transformed as follows:

\[ W_0\int_{-\infty}^{+\infty} e^{-\frac{a}{kT}(\lambda-\lambda_0)^2}\,d\lambda = W_0\sqrt{\frac{kT}{a}}\int_{-\infty}^{+\infty} e^{-\frac{a}{kT}(\lambda-\lambda_0)^2} \,d\left[\sqrt{\frac{a}{kT}}(\lambda-\lambda_0)\right] = \]

\[ = W_0\sqrt{\frac{kT}{a}}\cdot \int_{-\infty}^{+\infty} e^{-\xi^2}\,d\xi, \]

where it is denoted that

\[ \sqrt{\frac{a}{kT}}(\lambda-\lambda_0)=\xi; \]

the last integral, as is known from courses in analysis, is equal to \(\sqrt{\pi}\), whence follows the result given in the text.

degrees of freedom of the entire system under consideration as a whole must on the average be equal to \(\frac{kT}{2}\).

3. If the electric charges present in conductors are subject to thermal motion, then from the preceding exposition it follows that fluctuations must also occur in their distribution inside the conductor. To make the picture more concrete, let us assume that the positive ions in the conductor are firmly bound into a definite lattice, in which each ion occupies a definite place, while the negative charges move between them more or less freely. Then it is easy to see that in separate short intervals of time, in each small volume inside the conductor, the number of negative charges present will not be constant.

Fig. 2.

It may become now greater, now smaller than the number of positive charges; correspondingly, the charge and potential of the volume under consideration will fluctuate both in magnitude and in sign. Inside the conductor, spontaneous differences of potential must continually arise; the “microscopic” currents caused by them will equalize these differences of potential. Let us imagine a circuit possessing a definite capacitance \(C\), self-inductance \(L\), and ohmic resistance \(r\). Such a circuit does not remain indifferent to the local fluctuations of current and voltage occurring in it. The potential difference and the current strength in the whole circuit will likewise be subject to fluctuations. If no external electromotive force is applied to the circuit, then the current strength in it will on the average be equal to zero. But at particular moments currents of fluctuation origin will flow in it and corresponding potential differences will arise. Einstein first pointed this out in 1907.

To explain this idea, let us consider a simple mechanical analogy. Take a pendulum capable of oscillating only in one plane. If the pendulum is pushed, it will perform damped oscillations. The energy initially imparted to it will decrease owing to the resistance of the air. According to the equation of damped oscillation, its amplitude should in the course of time become arbitrarily small. In reality, however, this will not happen. Impacts of individual molecules will impart new oscillations to the pendulum, which in turn will die out, be renewed again, and so on. After the initial motion has died out, the pendulum will remain in a kind of Brownian motion maintained by the molecules of the surrounding air. The theory of this motion was also

given by Einstein and Smoluchowski. Applying the law of uniform distribution of energy over degrees of freedom, one may write that the mean kinetic energy of the pendulum is

\[ \frac{M\overline{v^{2}}}{2}=\frac{kT}{2}. \]

The mean potential energy in oscillatory motion is equal to the mean kinetic energy; therefore

\[ \overline{E}_{pot}=\frac{kT}{2}, \]

and the total energy of the pendulum is

\[ \overline{E}=\overline{E}_{kin}+\overline{E}_{pot}=kT. \]

In this example the dual role played by the thermal motion of air molecules is very clearly seen. It creates 1) friction, causing the damping of motions, 2) impulses that renew the motion. This entire reasoning can be transferred completely to fluctuations in an electrical circuit.

The parameters determining the state of a circuit are the current strength \(i\) and the voltage \(v\). On the basis of what was set forth in § [[unclear: section number not visible]], one may expect that, after all oscillations caused by external electromotive forces have ceased, there will remain in the circuit oscillations caused by thermal motion. The energy \(i\) must be such that

\[ \frac{L\overline{i^{2}}}{2}=\frac{kT}{2} \tag{[[unclear: formula number partly cut off]]} \]

and

\[ \frac{C\overline{v^{2}}}{2}=\frac{kT}{2}. \tag{[[unclear: formula number partly cut off]]} \]

It is interesting to estimate the order of magnitude of these currents and voltages. The quantity \(\frac{kT}{2}\) at the temperature \(17^\circ\mathrm{C}=290^\circ\mathrm{K}\) is equal to \(2.0\cdot 10^{-14}\) erg. Assuming, as Einstein did, \(C=5000\) c., we find:

\[ v_{mean}^{*}=\sqrt{\overline{v^{2}}} =\sqrt{\frac{kT}{C}} =\sqrt{\frac{4.0\cdot 10^{-14}}{5\cdot 10^{3}}} =2.8\cdot 10^{-9}\ \text{CGS units} =8.4\cdot 10^{-7}\ \mathrm{V}. \]

For \(L=0.1\) henry \(=10\) cm we find:

\[ i_{mean}=\sqrt{\overline{i^{2}}} =\sqrt{\frac{4.0\cdot 10^{-14}}{10^{8}}} =2\cdot 10^{-11}\ \text{weber} =2\cdot 10^{-10}\ \mathrm{A}. \]

Such are the mean values of the electrical fluctuations for the given \(C\) and \(L\). The true values of \(i\) and \(v\) may be smaller

* By “mean” \(i\), \(v\), etc., in this article we shall always mean root-mean-square values.

and greater than these means; the probability of each separate value is determined by formula (3″), in which instead of \(a\) one must insert \(\dfrac{L}{2}\) or \(\dfrac{C}{2}\), while \(W_0\) is determined from the condition

\[ \int_{-\infty}^{+\infty} W(\lambda)\, d\lambda = 1, \]

i.e., by formula (3‴).

Let us note that the expression for the mean energy of electrical fluctuations contains no quantities characterizing the circuit at all: this expression contains only the temperature and the universal constant \(k = \dfrac{R}{N}\). Proceeding from this, Einstein, in the same note of 1907,² proposed using the observation of electrical fluctuations to determine Avogadro’s number \(N\). The experiment was to be carried out according to the following scheme. A capacitor of capacitance of the order indicated above (\(5000\ \text{cm}\)) is short-circuited and then opened. At the moment of opening, there remained on it a certain voltage, the mean value of which we have already calculated. If, after opening, the plates of the capacitor are separated so that the capacitance decreases, for example to \(10\ \text{cm}\), then the voltage between them increases to:

\[ v_1 = 8 \cdot 10^{-7} \cdot \frac{5000}{10} = 4 \cdot 10^{-4}\ \mathrm{V}. \]

Such a potential difference can be detected with the aid of the most sensitive electrometers. By repeating the experiment a sufficiently large number of times, one can determine \(k\) and \(N\) from the mean result.

In this form, this proposal, so far as is known to the author of the present article, was not realized. As Einstein points out, it is difficult to carry out the make-and-break operations required by his proposal without at the same time producing extraneous electromotive forces of the same or a higher order, which would obscure the phenomenon sought. Below we shall describe a number of works that proved the existence of electrical fluctuations of thermal origin, but by entirely different methods.

1. The theoretical investigation of electrical fluctuations, begun by Einstein, was continued by G. L. de Haas-Lorentz³ in an interesting monograph devoted to Brownian motion. This investigator examined in greater detail fluctuations in a circuit with one degree of freedom and analyzed a number of new cases; we now turn to the presentation of her results.

Let the circuit we are studying consist of a conductor with self-inductance \(L\) and resistance \(r\). Such a circuit can possess only magnetic energy, which on the average is equal to

\[ \frac{L\overline{i^2}}{2} = \frac{kT}{2}. \]

The mean current strength \(\sqrt{\overline{i^2}}\) is determined by the formula

\[ \sqrt{\overline{i^2}}=\sqrt{\frac{kT}{L}} . \tag{5'} \]

Owing to the presence of resistance, every current that arises in this circuit must die out according to the equation

\[ i=i_0 e^{-\frac{r}{L}t}. \]

In order to maintain the current at a definite level, an electromotive force must be applied to the circuit. Therefore, phenomenologically, we may represent the action of thermal motion as a certain rapidly varying electromotive force, which we shall denote by \(E\). Then the equation for the current strength in this circuit may be written as

\[ L\frac{di}{dt}+ri=E. \tag{7} \]

Using equation (7) and the law of equipartition of energy, one can obtain certain information about the order of magnitude and the properties of the electromotive force \(E\). de Haas Lorentz uses for this derivation a method developed by Einstein and Hopf.^4 Let us integrate equation (7) over a time interval \(\tau\) so small that \(i\) does not change appreciably. Then we find:

\[ L(i_2-i_1)+ri_1\tau=\int E\,dt,\quad \text{or}\quad Li_2=(L-r\tau)i_1+\int E\,dt. \]

Let us square the last equality and take the mean, over a sufficiently large time interval, of all the terms entering into it:

\[ L^2\overline{i_2^2}=(L-r\tau)^2\overline{i_1^2} +2(L-r\tau)\,\overline{i_1\int E\,dt} +\overline{\left\{\int E\,dt\right\}^2}. \]

The magnitude and sign of the e.m.f. \(E\) do not in any way depend on the preceding current strength \(i_1\); on the average, they can equally often be of the same sign and of different signs. Therefore, over a sufficiently long time,

\[ \overline{i_1\int E\,dt}=0 \]

and

\[ L^2\overline{i_2^2}=(L-r\tau)^2\overline{i_1^2}+\overline{X^2}, \]

where the letter \(X\) denotes \(\int E\,dt\)—the “impulse” of the electromotive force over the time \(\tau\). Since on the average it must be

\[ \overline{i_1^2}=\overline{i_2^2}=\overline{i^2}, \]

it follows from equality (8) that

\[ \overline{X^2}=L^2\overline{i^2}-(L-r\tau)^2\overline{i^2} =(2rL\tau-r^2\tau^2)\overline{i^2}. \]

Let us choose the time interval \(\tau\ll \frac{L}{r}\), i.e. much smaller than the time during which the current strength in the given circuit already

increases by a factor of \(l\). Then in the last bracket the second term may be neglected. Since

\[ \overline{Li^2}=kT, \]

we finally find:

\[ \overline{X^2}=2rkT\tau . \tag{9} \]

This important formula gives an idea of the character and magnitude of the “electromotive impulses” that cause current fluctuations. Their mean square is proportional to the time and to the ohmic resistance of the circuit; the inductive resistance plays no role. The proportionality to the first power of the time can be explained by the following argument. Let us consider two successive intervals of time \(\tau_1\) and \(\tau_2\). The total impulse \(X\) will be equal to the algebraic sum of the impulses over both intervals:

\[ X=X_1+X_2 . \]

Squaring and taking the time average, we get:

\[ \overline{X^2}=\overline{X_1^2}+2\overline{X_1X_2}+\overline{X_2^2}. \]

In view of the complete independence of successive impulses from one another,

\[ \overline{X_1X_2}=0 . \]

and, consequently,

\[ \overline{X^2}=\overline{X_1^2}+\overline{X_2^2}, \]

whence it is seen that the square of the impulse over the time \(2\tau\) is twice as large as over the time \(\tau\).

By an analogous argument one can show that \(\overline{X^2}\) must be proportional to the first power of \(r\). For this it is necessary to take two successively connected pieces of wire of the same length and resistance and consider the total impulse in them. In essence, the connection between \(\overline{X^2}\) and \(r\) (and only \(r\), not \(L\)) follows from the fact that both are the result of thermal motion.

Let us pass to the case of two coupled circuits having resistances \(r_1, r_2\) and self-induction coefficients \(L_1, L_2\). Let the coefficient of mutual induction between them be \(M\). The equations satisfied by the currents in both circuits will have the form:

\[ L_1\frac{di_1}{dt}+M\frac{di_2}{dt}=-i_1r_1+E_1, \]

\[ L_2\frac{di_2}{dt}+M\frac{di_1}{dt}=-i_2r_2+E_2. \]

Analogously to how this was done for one circuit, we integrate these equalities over a small interval of time. We find:

\[ L_1(i_1'-i_1)+M(i_2'-i_2)=-i_1r_1\tau+X_1, \]

or

\[ L_1 i'_1 + M i'_2 = L_1 i_1 + M i_2 - i_1 r_1 \tau + X_1, \]

and the corresponding equality for the second circuit. Let us square this equality and take the mean over a sufficiently long interval of time:

\[ \overline{(L_1 i'_1 + M i'_2)^2} = \overline{(L_1 i_1 + M i_2)^2} - \overline{2 i_1 r_1 \tau (L_1 i_1 + M i_2)} + \overline{X_1^2}. \]

The term containing \(\tau^2\) we discard because of its smallness, while the remaining terms vanish upon averaging. Since

\[ \overline{(L_1 i'_1 + M i'_2)^2} = \overline{(L_1 i_1 + M i_2)^2}, \]

from the preceding equality there remains:

\[ \overline{X_1^2} - 2 i_1 r_1 \tau \, \overline{L_1 i_1 + M i_2} = 0, \]

or, taking into account that [see formula (9)]

\[ \overline{X_1^2} = 2 r_1 k T \tau, \]

we obtain:

\[ \overline{i_1 (L_1 i_1 + M i_2)} = kT. \]

In the same way, from the equation for the second circuit we find that

\[ \overline{i_2 (L_2 i_2 + M i_1)} = kT. \]

Dividing by 2 and adding the last two equalities, we see that

\[ \overline{\frac{1}{2} L_1 i_1^2 + M i_1 i_2 + \frac{1}{2} L_2 i_2^2} = kT. \tag{10} \]

This result proves that the mean energy of two coupled circuits is equal to the kinetic energy corresponding to two degrees of freedom.

De Haas further proves that the mean energy of an electric circuit does not depend on the particular features of its construction, if only the number of its degrees of freedom remains unchanged. Thus, for example, he considers a circuit composed of two different metals. In such a circuit, temperature differences may arise spontaneously between the two junctions, as was shown in one of Einstein’s early papers.^5 These temperature differences give rise to thermoelectric currents in the circuit. The mean value of the square of the current is found to be, all the same,

\[ \overline{i^2} = \frac{kT}{L} \]

and the mean magnetic energy in the circuit

\[ \frac{1}{2} L \overline{i^2} = \frac{1}{2} kT. \]

5. Finally, we shall present in full the analysis of one more question, which is of immediate practical

interest for the technique of electrical measurements. Namely, let us consider the motion of a magnetic needle, suspended at the center of a circular conductor, under the action of the fluctuating currents arising in the latter. We introduce the following notation: \(R\) is the radius of the wire, \(r\) is the resistance of the wire, \(i\) is the current in it, \(H\) is the intensity of the magnetic field produced by it at the center of the circle \(\left(H=\dfrac{2\pi i}{R}\right)\), \(M\) is the magnetic moment of the needle, \(Q\) is the moment of inertia of the needle, \(\vartheta\) is the angle of deflection of the needle from the plane of the conductor.

First we shall restrict ourselves to the case of a needle placed in a vacuum and not held in its equilibrium position by any elastic forces. Then the Brownian motion of the needle will be caused only by impulses from the fluctuating currents in the conductor. If one assumes that the needle is very small in comparison with \(R\) and that, consequently, the field in which it moves may be regarded as uniform, then the torque rotating it is expressed as follows:

\[ P = HM\cos\vartheta = \frac{2\pi i}{R}\,M\cos\vartheta \simeq \frac{2\pi iM}{R}, \tag{11} \]

if we restrict ourselves only to the consideration of very small deflections.

In its motion the needle will be retarded by induction currents arising in the circular conductor during its rotation. The strength of these currents is

\[ i_{\mathrm{ind}}=\frac{E_{\mathrm{ind}}}{r}=-\frac{1}{r}\frac{dN}{dt}, \]

where \(N\) is the magnetic flux of the needle penetrating the contour of the conductor. We shall determine it by using the theorem on the equivalence of a magnet and a current. The needle under consideration, with magnetic moment \(M\), may be replaced by a coil with number of turns \(n\) and cross-sectional area \(O\), with such a current \(j\) flowing through the coil that

\[ nOj=M. \]

Further, the flux created by the current \(j\) in the coil and penetrating the contour of the circular conductor is equal to the flux that would be produced by a current of the same strength \(j\) in the circular conductor and would penetrate the cross-section of the coil. The latter may be written as

\[ N=\frac{2\pi j}{R}\,On\cdot\sin\vartheta=\frac{2\pi M}{R}\sin\vartheta. \]

Consequently,

\[ i_{\mathrm{ind}}=-\frac{2\pi M}{Rr}\cos\vartheta\,\frac{d\vartheta}{dt}\simeq -\frac{2\pi M}{Rr}\frac{d\vartheta}{dt}, \]

where again \(\cos\vartheta\simeq 1\) has been put.

The retarding torque acting on the needle will be expressed by the following formula:

\[ P_s=\frac{2\pi i_{\mathrm{ind}}}{R}M=-\frac{4\pi^2 M^2}{R^2 r}\frac{d\vartheta}{dt}. \]

Now one can write the equation of motion of the pointer. Denoting \(\dfrac{d\varphi}{dt}=\omega\), we shall write, neglecting the action of the earth’s magnetic field on the pointer,*

\[ Q\frac{d\omega}{dt}=\frac{2\pi iM}{R}-\frac{4\pi^2M^2}{R^2r}\,\omega . \tag{12} \]

This equation is of the same form as equation (7) for the current in an insulated circuit. It can be studied in the same way, by applying the Einstein–Hopf method.

Let us integrate it over a small time \(\tau\), such that \(\omega\) changes very little. We find:

\[ Q(\omega'-\omega)=\frac{2\pi M}{R}\int_{(\tau)} i\,dt-\frac{4\pi^2M^2}{R^2r}\,\omega\tau; \]

taking into account that \(\int i\,dt=e\)—the quantity of electricity flowing during the time \(\tau\)—we rewrite the result of the integration in the form:

\[ Q\omega'=\left[Q-\frac{4\pi^2M^2}{R^2r}\,\tau\right]\omega+\frac{2\pi M}{R}\,e. \]

Squaring and taking the mean over a long interval of time:

\[ Q^2\overline{\omega'^2}=\left[Q^2-\frac{8\pi^2M^2}{R^2r}\,Q\tau\right]\overline{\omega^2} +\frac{4\pi^2M^2}{R^2}\,\overline{e^2}. \tag{13} \]

As before, here the term with \(\tau^2\) has been omitted because of its smallness, and the term containing \(e\omega\), since the latter quantity is equal to zero. For \(\overline{e^2}\), de Haas–Lorentz, using the analogy with Einstein’s formula determining the square of the mean displacement of a Brownian particle in some definite direction, gives the expression:

\[ \overline{e^2}=\frac{2kT}{r}\,\tau . \]

Substituting this expression into formula (13), noting that \(\overline{\omega'^2}=\overline{\omega^2}\), and canceling the corresponding terms, after simplification we find:

\[ \overline{Q\omega^2}=kT. \]

Consequently, in this case as well, the mean kinetic energy of the pointer, which is a body with one degree of freedom, turns out to be equal to:

\[ \frac{1}{2}\,Q\,\overline{\omega^2}=\frac{1}{2}\,kT. \]

Now let us place the whole system in air and consider how the presence of the latter will affect the motion of the pointer. Still—

* This can be done if the whole instrument is placed in an iron armor of sufficient thickness, or if the earth’s magnetic field is compensated by a counter-field.

we shall suppose that there is no directive force, such as that of the earth’s field, the elasticity of the filament, etc. Then two additional terms will enter into the equation of motion of the pointer (12); one of them, equal to \(-w\omega\), will represent the resistance of the air; the other, the impulses received by the pointer from the gas molecules; we shall denote it by \(F\):

\[ Q\frac{d\omega}{dt}=\frac{2\pi iM}{R}+F-\frac{4\pi^2M^2}{R^2r}\,\omega-w\omega . \tag{12'} \]

We proceed in the same way as in the preceding case: we integrate over the time \(\tau\) and collect like terms:

\[ Q\omega'=\left(Q-w\tau-\frac{4\pi^2M^2}{R^2r}\,\tau\right)\omega+\frac{2\pi M}{R}\,e+Y, \]

where

\[ Y=\int_{(\tau)} Fdt. \]

We square and average over a large interval of time; taking into account that \(\overline{(\omega')^2}=\overline{\omega^2}_1\), \(\overline{Ye}=0\), \(\overline{Y\omega}=0\) and \(\overline{\omega e}=0\), and discarding the term with \(\tau^2\), we find:

\[ 2Q\left(w\tau+\frac{4\pi^2M^2}{R^2r}\,\tau\right)\overline{\omega^2} = \frac{4\pi^2M^2}{R^2}\overline{e^2}+\overline{Y^2}. \tag{13'} \]

The quantity \(\overline{e^2}\), as has already been indicated, is expressed as follows:

\[ \overline{e^2}=\frac{2kT}{r}\,\tau. \]

As for \(\overline{Y^2}\), this quantity is obtained from consideration of the Brownian motion of the pointer in air in the absence of a conductor. The law of uniform distribution of energy gives in this case, for \(\overline{Y^2}\), the expression

\[ \overline{Y^2}=2kTw\tau, \]

which is quite analogous to expression (9) for \(\overline{X^2}\) in a closed circuit. Substituting the values of \(\overline{e^2}\) and \(\overline{Y^2}\) in (13′), and reducing by

\[ 2\left(w\tau+\frac{4\pi^2M^2}{R^2r}\,\tau\right); \]

there remains

\[ Q\overline{\omega^2}=kT. \]

Consequently, for the kinetic energy \(\frac{1}{2}Q\overline{\omega^2}\) there remains in force the expression derived for the case of a pointer in vacuum.

This remarkable result shows that whatever the mechanism of transmission of molecular motion to a given body, the energy of motion remains the same on the average. Several different factors together produce no greater fluctuations than each of them separately. The reason for this is clearly visible in the course of the preceding derivation. Each process capable of produc-

vate the Brownian motion of a given system plays a double role in relation to it: on the one hand, it excites its motion by disorderly impulses; on the other hand, it retards its motion by its resistance. When several factors act simultaneously, their impulses are added together on the one hand, and their resistances on the other. Therefore the resultant motion is neither strengthened nor weakened. This circumstance can easily be illustrated by the fact that the energy of Brownian motion does not depend on the pressure of the surrounding gas, i.e. on the frequency of impacts of the molecules. The more often the molecules strike, the greater the impulses they transmit per unit time to the oscillating system; however, the greater also is the resistance which they offer to its motion.

It does not follow from this, however, that the magnitude of the resistance plays no role. It affects the character of the motion, its greater or lesser regularity. At small resistances, and consequently rare jolts, a system undergoing Brownian motion moves continuously for a longer time; as the resistance increases, the number of discontinuities in velocity per unit time increases. A pendulum hanging in atmospheric air performs very disorderly motions, in which there is almost no trace of its own period. Conversely, in a high vacuum its motion would consist of considerable segments of a complete oscillation and even whole oscillations, the initial phase of which changes abruptly from time to time.

In the case of electrical oscillations, at small resistance we should have an almost pure natural tone of the circuit; if such fluctuations were amplified and observed with the aid of a telephone, we would hear a musical tone. With a large resistance of the circuit, the fluctuations in it would be heard as a rustle. This circumstance was noted by W. Schottky.^6

Thermal Motion and the Limit of Sensitivity of Galvanometers

6. The discovery and study of electrical fluctuations has not only theoretical significance. The development of the technique of weak electric currents in recent decades has gradually introduced ever weaker currents and voltages into the field of activity of the experimental and practical electrician. Even earlier in laboratory practice it was necessary to deal with measurements of extremely small currents; this need was met more or less satisfactorily by a number of developed systems of galvanometers and electrometers. The invention of the cathode tube and the creation of multi-tube amplifiers, allowing enormous amplification in voltage, current, and power, placed in the hands of electricians new and much richer means of working with weak currents.

The development of radio, the introduction into technology of photoelectric cells, new forms of communication, sound cinema, television, etc., made currents of the order of \(10^{-8}—10^{-10}\ \mathrm{A}\) and voltages of the order of \(10^{-3}—10^{-4}\ \mathrm{V}\) commonplace objects of technology. The widespread dissemination of weak-current technology created increased demands for the accuracy and simplicity of electrical measuring weak-current apparatus; the old armored galvanometers, with their heavy protection, complex and inconvenient system of astasis, and enormous periods, which greatly lengthened the observation time, could no longer satisfy modern requirements. Design thinking responded to these requirements with new constructions, such as, for example, paired galvanometers of the Moll and Burger thermorelay type1 and its numerous variants, which far surpassed the old instruments in sensitivity while at the same time being considerably more accessible and simpler to use. Naturally, the intensification of work in the field of extremely weak currents forced experimenters to pay attention to the existence of electrical fluctuations. This question was approached almost simultaneously from two sides: on the one hand, by studying the limiting possibilities of galvanometers, and on the other, by analyzing the disturbances existing in the operation of tube receivers and amplifiers. It is curious to note that between the works of Einstein and de Haas-Lorentz, which laid the foundations of the theory of electrical fluctuations, and the transfer of the question of fluctuations into the practical plane, almost ten years passed; it was precisely during this time that the cathode tube was discovered and mastered.

The significance of electrical fluctuations in weak-current technology consists in the fact that they set a definite limit for the sensitivity* of electrical instruments. This applies both to measuring instruments (galvanometers, electrometers) and to receiving and amplifying devices. In fact, in any conductor, according to the theory set forth above, spontaneous motions of electricity occur, and microcurrents and microvoltages arise. Suppose that in some circuit a current must flow whose magnitude is to be measured; simplifying the situation somewhat, one may say that this is feasible only if the strength of the current being measured exceeds the mean strength of the fluctuation currents. But if the current to be measured is weaker than the currents arising from thermal motion, then it cannot be measured, even if the sensitivity of the galvanometer is sufficient for this; in this case the measured current

will be masked by fluctuations (as we shall see below, in this form the statement is not sufficiently precise; a correction must be introduced into it).

The same applies, in the main, to the operation of receiving devices. If the signal received in a radio receiver is weaker than the effect of thermal motion, then we shall not be able to distinguish it; it will be covered by fluctuation currents. Therefore the study of electrical fluctuations must provide practical electrical engineering and electrometry with indications of the limits of observable electrical quantities, of the factors on which these limits depend, and of what must be done in order, as far as possible, to lower them.

1. Let us begin the exposition of the question by considering the limit of sensitivity of galvanometers. The basic fact was revealed with complete clarity in the work of de Haas-Lorentz in the part concerning the Brownian motion of a magnetic needle inside a ring conductor. The case considered by her is nothing other than a tangent compass, the prototype of a galvanometer with a moving magnet, in the discussion of which, for simplicity, the directive forces were omitted. We saw that the needle of this galvanometer must be in Brownian motion, whose mean kinetic energy is

\[ \frac{1}{2} Q\overline{\omega^{2}}=\frac{1}{2}kT. \]

In fact, when considering the motion of the needle of a galvanometer, the directive force cannot be discarded, since almost all accepted designs of electrical measuring instruments could not serve for measurement without this force; it determines both the zero point and the sensitivity of the instrument. This means that, besides kinetic energy, the moving system in a galvanometer must also possess potential energy. If the angle of deflection is denoted by \(\varphi\), and the directive force by \(f=-\Delta\varphi\), then the mean potential energy of the magnetic system will be:

\[ \frac{1}{2}\Delta\overline{\varphi^{2}}=\frac{1}{2}kT. \tag{14} \]

Hence we find

\[ \varphi_{\text{mean}}=\sqrt{\overline{\varphi^{2}}}=\sqrt{\frac{kT}{\Delta}}; \]

such is the mean deflection to which the needle will be subjected as a result of fluctuation jolts. Let us recall that this quantity does not depend on whether the needle is in a vacuum or in an atmosphere of some gas, i.e. on whether only the currents flowing in the conductor act upon it, or also the molecules of the gas. The same oscillations will also occur in a galvanometer with a moving coil. The conclusion of de Haas-Lorentz must here be reversed. Fluctuations of current will occur in the moving-

coil. In the field of stationary magnets it will experience impulses that will not allow it to remain at rest. To these impulses there will also be added the jolts of air molecules; this circumstance will not affect the amplitude of the coil’s oscillations. The latter will remain even in the case when the galvanometer coil is open and current fluctuations do not occur. In this case the oscillations of the coil will occur entirely because of the motion of the air molecules. Ising^8,11 was the first to point out these phenomena as the practical limit of the sensitivity of a galvanometer.

His work was continued by Czerny^12, Ornstein and collaborators^13,14, and Czerny^15. As a result we have a clear picture, although one far from completed in all its details, describing the role of fluctuations in galvanometry.

First of all let us find out what errors arise from thermal motion in a single measurement of the current strength. Let the measured current strength \(\delta i\) cause a deflection of the galvanometer by an angle \(\delta \varphi\). If \(\delta \varphi\) is less than \(\varphi_{\text{avg}}\), determined by formula (14), then it is clear that the galvanometer reading will be illusory. In a single reading we shall not be able to say whether the observed deflection is produced by the measured current or by fluctuations. In order to have any confidence in the value of the observed deflection, it must be greater than \(\varphi_{\text{avg}}\); moreover, the larger \(\delta \varphi\), the greater the reliability of the result. To judge how great the probability is of taking a fluctuation deflection for the measured quantity, we give the following table, borrowed from the article by M. Czerny.

TABLE 1

Deflection	Energy of the deflected coil	Probability of fluctuation
\(\geq 1\cdot \varphi_{\text{avg}}\)	\(\geq 1\cdot \dfrac{1}{2} kT\)	0.317
\(\geq 2\cdot \varphi_{\text{avg}}\)	\(\geq 4\cdot \dfrac{1}{2} kT\)	0.045
\(\geq 3\cdot \varphi_{\text{avg}}\)	\(\geq 9\cdot \dfrac{1}{2} kT\)	0.003
\(\geq 4\cdot \varphi_{\text{avg}}\)	\(\geq 16\cdot \dfrac{1}{2} kT\)	0.00006

The quantities appearing in the last column are calculated from formula (3′); namely, the probability that the spontaneous deflection will be equal to or greater than \(\varphi\) is expressed as follows:

\[ W=\int_{\varphi}^{\infty} W(\varphi)\,d\varphi = \int_{\varphi}^{\infty} \frac{1}{\sqrt{2\pi \varphi^{2}}}\, e^{-\frac{4\alpha^{2}}{kT}}\,d\varphi . \]

The table shows that for deflections 3–4 times greater than the magnitude of the fluctuations, the probability of a completely erroneous reading is very small. If, however, we determine them

to the galvanometer currents that cause deflections of the order of \(\varphi_{\mathrm{avg}}\), then in one case out of three we shall take a random deflection of the galvanometer for the action of the current being measured. For \(\varphi < \varphi_{\mathrm{avg}}\) the probability of error is still greater. Therefore Ising considers reliable only readings \(\geq 4\varphi_{\mathrm{avg}}\). With such deflections one may practically have no doubt that they are caused by an actual extraneous current. However, this does not mean that the magnitude of the reading obtained gives the magnitude of the current with an accuracy equal to \(0.00006\). The probable error of the reading may, in a first approximation, be determined as follows. The mean absolute value of the spontaneous deflection of the galvanometer is

\[ |\overline{\varphi}| = 2 \int_{0}^{\infty} \varphi W(\varphi)\,d\varphi = \frac{2}{\sqrt{2\pi\varphi^{2}}} \int_{0}^{\infty} \varphi e^{-\frac{\Delta \varphi^{2}}{kT}}\,d\varphi = \sqrt{\frac{2kT}{\pi\Delta}} = \sqrt{\frac{2}{\pi}}\,\varphi_{\mathrm{avg}}. \]

This quantity is the probable error of a single measurement. If the deflection is \(\varphi = 4\varphi_{\mathrm{avg}}\), then the probable relative error of the measurement will be:

\[ \sqrt{\frac{2}{\pi}\frac{\varphi_{\mathrm{avg}}}{4\varphi_{\mathrm{avg}}}} \simeq 0.20. \]

But this result cannot be regarded as rigorous. In an exact treatment, the error of an individual observation must be taken to be the difference of the spontaneous deflections of the galvanometer at the beginning and at the end of the measurement. These deflections are not entirely independent of one another. The magnitude of the final deflection depends on the magnitude of the initial deflection; it is more probable that at the end of the measurement there will be deflections not very far from the initial one, especially if the time interval occupied by the measurement is not very large. This kind of influence of the preceding state on the probability of the subsequent one was considered by M. Smoluchowski in the theory of Brownian motion; Smoluchowski called it “probabilistic aftereffect” (Wahrscheinlichkeitsnachwirkung). Such a quantity must also enter into the exact theory of the Brownian motion of a galvanometer; however, this work has not yet been done up to the present time (1933).

Let us now proceed to the determination of those currents and voltages to which the above-described limit of measurement accuracy corresponds.

For this it is necessary, besides the already determined limit of accuracy of the directly read quantity (for example, the angle of deflection), to know the sensitivity of the galvanometer to current and voltage, expressed in units of this quantity. Ising does this in the following way.

The moving part of the galvanometer, through which a current \(\delta i\) flows, experiences the action of a couple, whose moment is equal to \(B\delta i\); the quantity \(B\) we shall call the “dynamic constant” of the galvanometer. Let

the deflection of the galvanometer is equal to \(\delta x\), where \(x\) is a quantity of any dimension—the angle of deflection, the displacement of the “spot” on the scale, the number of divisions, etc.; the opposing moment developed by the galvanometer at such a deflection is equal to \(-A\delta x\), where \(A\) is the generalized force for \(\delta x=1\), usually called the “directive force.” The equilibrium condition is:

\[ A\delta x=B\delta i . \tag{15} \]

The sensitivity of the galvanometer is defined by the equality

\[ S_i=\frac{\delta x}{\delta i} \tag{16} \]

(“current” sensitivity); or

\[ S_v=\frac{\delta x}{\delta v} \]

(“voltage” sensitivity).

Determining \(\dfrac{\delta x}{\delta i}\) from equality (15), we find:

\[ S_i=\frac{B}{A}. \tag{16'} \]

For every type of instrument one can find a definite relation between \(B\), \(A\), and a series of directly measurable quantities. This relation, together with (16′) and the condition

\[ \frac{1}{2}A\overline{\delta x^2}=\frac{1}{2}kT \]

gives the desired limiting value of the sensitivity. Let us write the frequency of the coil’s natural undamped oscillations:

\[ \omega_0=\sqrt{\frac{A}{K}}, \]

where \(K\) is the moment of inertia.

Further, the damping coefficient of the coil oscillations is:

\[ \lambda=\frac{p_0}{2K}+\frac{B^2}{2KR}, \]

where \(p_0\) is the coefficient of friction of the coil in air, and \(R\) is its ohmic resistance.* If \(R\) is not very large, then the second term

* This equality is derived as follows. The braking moment acting on the coil is made up of the resistance of the air, \(-p_0\dot{\varphi}\), and of the electrodynamic couple of forces due to the induction current in the coil. The latter is equal to:

\[ i_{\mathrm{ind}}=-\frac{snH}{R}\frac{d\varphi}{dt}, \]

where \(s\) is the area of the coil, \(n\) is the number of turns, and \(H\) is the magnetic-field strength. The moment of the opposing couple is:

\[ M_{\mathrm{ind}}=snH\cdot i_{\mathrm{ind}} =-\frac{s^2n^2H^2}{R}\frac{d\varphi}{dt}. \]

Since \(snH=B\), we have \(M_{\mathrm{ind}}=-\dfrac{B^2}{R}\dot{\varphi}\), and the total braking moment is:

\[ M=-\left(p_0+\frac{B^2}{R}\right)\dot{\varphi}=-p_1\dot{\varphi}. \]

Further, \(\lambda=\dfrac{p_1}{2K}\), whence we obtain the formula given in the text.

in the expression for \(\lambda\), which represents electromagnetic damping, is usually large compared with the first, which gives the magnitude of the damping by air; therefore we shall discard the first term. Thus, as the cause of motion, we in fact take only the current fluctuations in the coil. Let us suppose that we are considering the case of critical damping, the most interesting one for practice, in which

\[ \omega_0=\lambda. \]

By virtue of the formulas written above this gives:

\[ \omega_0=\sqrt{\frac{A}{K}}=\frac{B^2}{2KR} \]

or

\[ B=\sqrt{2KR\omega_0}. \]

Substituting

\[ K=\frac{A}{\omega_0^2}, \]

we find

\[ B=\sqrt{\frac{2AR}{\omega_0}} \]

and

\[ s_i=\frac{B}{A}=\sqrt{\frac{2R}{A\omega_0}}. \]

The smallest reading that, according to Ising, possesses a certain reliability is

\[ x_{\min}=4x_{\text{mean}}=4\sqrt{\frac{kT}{A}}. \]

Hence the minimum current that can be reliably determined by means of the given galvanometer is

\[ i_{\min}=\frac{x_{\min}}{s_i} =4\sqrt{\frac{kT\omega_0}{2R}} =4\sqrt{\frac{\pi kT}{R\vartheta_0}}, \tag{17} \]

where \(\vartheta_0\) is the period of oscillation of the coil in the absence of friction.

This quantity is 4 times greater than the mean fluctuation current flowing through the coil of the galvanometer, the strength of which we can judge from the magnitude of the mean deflection:

\[ i_{\text{mean}}=\frac{x_{\text{mean}}}{s_i} =\sqrt{\frac{\pi kT}{R\vartheta_0}}. \]

From these formulas it is evident that there is no absolute limit for the measurable current strength. By increasing the resistance and the period of the galvanometer, we can obtain the possibility of measuring arbitrarily weak currents. But for a given galvanometer formula (17) characterizes the limiting accuracy of a single reading. The last reservation is of essential importance. If meas—

direct current does not change for a long time, and we make observations for so long that we can determine, with some reliability, the new position of the zero point as the average of many readings, then the error can be made arbitrarily small. The same result is obtained if we increase the period of the galvanometer without limit; in both cases the time required for a single reading increases without limit. From formula (17) it is evident that in this case the limit of the measurable current strength tends to zero. The second remark that must be made concerning formula (17) is that it assumes that the initial position of the zero point is known exactly. In practice one often proceeds otherwise, namely, the reading of the instrument is taken immediately before switching on and then after switching on and after the instrument has come to rest. In this case, as we have already indicated, the Wahrscheinlichkeitsnachwirkung must be taken into account. If, however, it is neglected, then as an approximation we obtain an average error \(\sqrt{2}\) times greater than with an exactly determined zero point.

The limit of accuracy of voltage measurement is easily obtained from formula (17):

\[ v_{\min}=i_{\min}R=4\sqrt{\frac{\pi kTR}{\vartheta_0}}=4v_{\text{avg}}. \tag{17'} \]

After what was said above about the limit for measuring current strength, formula (17′) hardly needs explanation. It is interesting to note that in both formulas the characteristics of the measuring instrument are represented only by its proper period and resistance; all other quantities, including sensitivity, have dropped out of the final result. The result obtained does not depend on the galvanometer system; in particular, it applies entirely

TABLE 2

\(R\) (ohm)	\(\vartheta_0=2\) sec. \(i_{\text{avg}}\) (A)	\(\vartheta_0=2\) sec. \(v_{\text{avg}}\) (V)	\(\vartheta_0=8\) sec. \(i_{\text{avg}}\) (A)	\(\vartheta_0=8\) sec. \(v_{\text{avg}}\) (V)
\(1\)	\(7.92\cdot10^{-11}\)	\(7.92\cdot10^{-10}\)	\(3.96\cdot10^{-11}\)	\(3.96\cdot10^{-11}\)
\(10\)	\(2.50\cdot10^{-11}\)	\(2.50\cdot10^{-10}\)	\(1.25\cdot10^{-11}\)	\(1.25\cdot10^{-10}\)
\(10^2\)	\(7.92\cdot10^{-12}\)	\(7.92\cdot10^{-10}\)	\(3.96\cdot10^{-12}\)	\(3.96\cdot10^{-10}\)
\(10^3\)	\(2.50\cdot10^{-12}\)	\(2.50\cdot10^{-9}\)	\(1.25\cdot10^{-12}\)	\(1.25\cdot10^{-9}\)
\(10^4\)	\(7.92\cdot10^{-13}\)	\(7.92\cdot10^{-9}\)	\(3.96\cdot10^{-13}\)	\(3.96\cdot10^{-9}\)
\(10^5\)	\(2.50\cdot10^{-13}\)	\(2.50\cdot10^{-8}\)	\(1.25\cdot10^{-13}\)	\(1.25\cdot10^{-8}\)
\(10^6\)	\(0.792\cdot10^{-13}\)	\(7.92\cdot10^{-8}\)	\(3.96\cdot10^{-14}\)	\(3.96\cdot10^{-8}\)

to galvanometers with moving magnets. To illustrate these formulas with concrete examples, we give Table 2 for \(i_{\text{avg}}\) and \(v_{\text{avg}}\), calculated by Ising for various \(\vartheta_0\) and \(R\). Let us recall that \(i_{\min}\) and \(v_{\min}\) are 4 times greater,

This table shows that fluctuations of electrical quantities have a magnitude already attained by the sensitivity of modern galvanometers. It is therefore natural to ask whether such fluctuations have not been observed in work with highly sensitive galvanometers. It turns out that investigators who constructed extremely sensitive galvanometers and radiomicrometers* long ago encountered the presence of incessant motions of the zero point. They were usually ascribed to the action of various external shocks, transmitted to the suspension through the walls of the building, the supports of the galvanometer, the air, etc. (so-called microseismic shocks). However, with the passage of time it became clear that these motions cannot be completely eliminated by any methods protecting the galvanometer from external shocks, such as: improved methods of suspending the galvanometer, increasing its mass, etc.; and that these motions occur in the same way in very different localities.¹² The method proposed by Moll and Burger for increasing the observed deflection by means of a differential thermoelement and a second galvanometer makes it possible to observe and study these motions with still greater clarity. In their work Moll and Burger⁷˒¹⁶ give a number of curves representing the recording of the motions of the coil of the first galvanometer at different magnifications. These curves are reproduced in Fig. 3. They clearly show both the disordered character of the motions and their constant (on the average) magnitude. Zernike subjected one of these curves (d, corresponding to a hundredfold amplification of the ther—

Fig. 3. Recording of the zero point and deflections of the galvanometer at different magnifications of the thermorelay.

* A combination of a radiation thermoelement with a galvanometer in one instrument: a coil consisting of two different wires, suspended on a fiber in a magnetic field.

** The reproduction of the original oscillograms (Figs. 3 and 5) has been replaced by schematic drawings.

to quantitative analysis. A section of this curve was taken; on section I the deflection was measured at 190 points, on section II at 390 points, and on section III at 105 points (685 points in all). For greater accuracy of measurement, Ising enlarged the scale of the curves photographically by a factor of 10.3. It turned out that the deflections had the following magnitude:

\[ \text{For section } I:\quad \sum x^2 = 778.1\ \mathrm{mm}^2,\quad \overline{x^2}=\frac{778.1}{190}=4.10\ \mathrm{mm}^2, \]

\[ \text{For section } II:\quad \sum x^2 = 1524.6\ \mathrm{mm}^2,\quad \overline{x^2}=\frac{1524.6}{390}=3.91\ \mathrm{mm}^2, \]

\[ \text{For section } III:\quad \sum x^2 = 343.5\ \mathrm{mm}^2,\quad \overline{x^2}=\frac{343.5}{105}=3.27\ \mathrm{mm}^2, \]

\[ \text{For all sections:}\quad \sum x^2 = 2646.6\ \mathrm{mm}^2,\quad \overline{x^2}=\frac{2646.2}{685}=3.86\ \mathrm{mm}^2, \]

whence

\[ x_{\mathrm{mean}}=\sqrt{\overline{x^2}}=1.97\ \mathrm{mm}. \]

As is seen from the data of Moll and Burger, a voltage of \(1\cdot 10^{-7}\ \mathrm{V}\) produced a deflection \(x=213.5\ \mathrm{mm}\); consequently, \(x_{\mathrm{mean}}=1.97\ \mathrm{mm}\) corresponds to \(v_{\mathrm{mean}}=9.2\cdot 10^{-10}\ \mathrm{V}\).* The period of the Ising galvanometer is determined, according to the authors’ data, as \(\vartheta_0=1.6\ \mathrm{sec}\), and its internal resistance is \(R=50\ \Omega\). Hence, using formula (17′), we find that, according to Ising’s theory, it should have been \(v_{\mathrm{mean}}=6.3\cdot 10^{-10}\ \mathrm{V}\). From comparison of the theoretical and observed values, Ising concludes that in the experiments of Moll and Burger the motions of the coil were for the most part (by \(2/3\)) caused by Brownian motion, and that only a small part was due to extraneous (microseismic) shocks. In fact, owing to the inaccuracy of the calibration, the fluctuation value derived by Ising from the data of Moll and Burger is smaller than the true one; but the conclusion he drew is qualitatively correct.

V. Einthoven and co-workers\(^{17}\) likewise observed Brownian motion while investigating a string galvanometer with an extremely thin thread. They established that spontaneous oscillations of the string do indeed arise from fluctuation impacts. The degree of damping does not affect the magnitude of the thread’s fluctuations, but it strongly affects the character of their course in time, in the sense indicated on p. 819: the smaller the damping, the more the string’s own period makes itself felt. This last observation is a confirmation of the law of equipartition of energy as applied to the type of fluctuations under consideration.

* Calibration of the readings of a thermorelay, obtained during rapid motions of the primary galvanometer by comparison with prolonged constant deflections, is in general incorrect and entails a certain error (see further, p. 824).

F. Parnyake2 constructed an apparatus operating on the principle of the thermorelay of Moll and Burger, but considerably surpassing it in sensitivity. Without giving any details, the author states that an essential part of it is a differential thermopile connected to a galvanometer by the same author (manufactured by Kipp and Zonen, Holland). This device makes it possible to increase the deflection of the first galvanometer by 12,000 times, i.e., with a light path length of 50 cm, to have a sensitivity corresponding to a scale displacement of 6 km! In such an apparatus the amplitude of random oscillations reaches 60 mm. Subsequently Czerny, with another, less sensitive apparatus, confirmed Scheinchen’s results on the influence of the degree of damping on Brownian motion. A fluctuation amplitude equal to 3 mm did not change when the first galvanometer was short-circuited, nor when the positions of the first and second galvanometers were interchanged. But the course of the fluctuations changed sharply in time: with the galvanometer open, the light pointer each time executed several regular swings in succession; with it short-circuited, it moved irregularly back and forth.

Finally, last year (1932) Ising succeeded in observing the fluctuations of a galvanometer with a rotating coil, without resorting to amplification of its deflections by means of a thermorelay. In his striving toward the solution of this problem Ising was guided by the following idea: the galvanometer with ultimate sensitivity, limited only by fluctuations, can only be a galvanometer with a rotating coil. Galvanometers with movable magnets are too sensitive to external influences, especially to variations of the magnetic field; therefore their sensitivity threshold in practice lies considerably above the fluctuation level. As for galvanometers with movable coils, their sensitivity is usually insufficiently high; to increase it by means of a second galvanometer and a light relay generally introduces new sources of error. In its main features the construction of Ising’s galvanometer is as follows. A light circular coil (diameter 17.4 mm, 25 turns of wire of diameter 0.05 mm) is mounted on an axle consisting of a piece of thin glass tube. The coil is held by two quartz threads (length 5 mm, diameter 8 μ), stretched by means of springs. Such fastening of the coil, as it turned out, provides it with good stability against external shocks. For reading the angle of rotation of the coil, a piece of quartz thread (length 35 mm) is attached to it perpendicular to the axis of rotation and serves as a pointer. It is viewed in a microscope with a magnification of the order of 10,000 times for visual observation and 1,500–2,000 times for photographic recording. For this purpose the parts making up the galvanometer proper are placed directly on the object stage of the microscope. In order

increase the accuracy of reading, Ising introduces a central diaphragm into the path of the rays, and takes the light source in the form of a narrow, brightly illuminated slit. Then in the field of view (or on the screen, in the case of projection) a series of diffraction bands is clearly visible, which serve for reading the displacement. To increase still further the sensitivity of his instrument, Ising applies electrostatic astasia: the pointer is covered with a conducting layer, which is connected by means of a thin wire with one end of the coil winding; around the pointer an electric field is produced by two plates forming a condenser. The sign and magnitude of the field are chosen so as, if possible, to diminish the directing force acting on the coil. It turned out that, for normal operation of the galvanometer, this astasia is superfluous; Ising used it only for varying the directing force when studying Brownian motion.

TABLE 3

Method of observation	Directing force	Number of measured points	$\overline{|\varphi|}\cdot \sqrt{\frac{\pi}{2}}\cdot 10^7$	Remark
Visual	$0.142\ \dfrac{\mathrm{dyn}}{\mathrm{cm}}$	41	4.31
”	$0.142\ \text{”}$	30	6.70
”	$0.142\ \text{”}$	83	6.46
”	$0.142\ \text{”}$	$\dfrac{54}{208}$	$\dfrac{5.42}{5.81}$
Photographic	$0.142\ \text{”}$	62	7.98	in the external circuit normal resistance
”	$0.142\ \text{”}$	47	5.64	in the external circuit the coil is short-circuited
”	$0.142\ \text{”}$	47	6.11	short-circuited
”	$0.142\ \text{”}$	51	5.06	short-circuited; presence of disturbance
”	$0.142\ \text{”}$	103	9.85	in the external circuit the coil
”	$0.142\ \text{”}$	51	6.02	calculated value $16.8\cdot 10^7$
”	$\dfrac{0.142}{10.2}=0.0139\ \dfrac{\mathrm{dyn}}{\mathrm{cm}}$	50	16.6	calculated value $16.8\cdot 10^7$
”	$\dfrac{0.142}{4.0}=0.0356\ \dfrac{\mathrm{dyn}}{\mathrm{cm}}$	51	8.52	calculated value $10.6\cdot 10^7$

In the photographs obtained by Ising, fluctuations of the zero point of the galvanometer are indeed visible. Ising determined their magnitude by first constructing an averaged curve representing the smooth displacement of the galvanometer due to extraneous electromotive forces, and reading from it the deviations in one and the other

Thus numerous curves were processed, obtained both visually and photographically. In doing so the experimental conditions were varied: an external resistance was connected, the directing force was changed, etc. The results are given in Table 3; \(|\bar{\varphi}|\) is the mean absolute value of the deflection; multiplying it by \(\sqrt{\pi/2}\), we obtain \(\sqrt{\overline{\varphi^{2}}}\) (see above, p. 823).

The mean from observations at \(\Delta = 0.142 \dfrac{\text{dyne}}{\text{cm}}\) is

\[ |\bar{\varphi}|\sqrt{\frac{\pi}{2}} = 5.90 \cdot 10^{-7}; \]

introducing a correction for measurement errors (errors in setting the microscope on the maximum blackening, etc.), the author reduces this value to \(5.60 \cdot 10^{-7}\). The value calculated from the formula

\[ \frac{1}{2}\Delta \overline{x^{2}} = \frac{1}{2}kT \]

is

\[ \sqrt{\overline{x^{2}}} = 5.31 \cdot 10^{-7}; \]

the discrepancy is about 6%. At \(\Delta = 0.0139 \dfrac{\text{dyne}}{\text{cm}}\) the agreement proves to be very good, while at \(\Delta = 0.0356 \dfrac{\text{dyne}}{\text{cm}}\) it is rather poor.

In the end, despite the fairly large scatter of the points, it must be acknowledged that the results of these measurements confirm the theory based on the law of equipartition of energy. To characterize the accuracy of the results obtained, one may try to determine from them Boltzmann’s constant \(k\) by the formula

\[ k=\frac{\Delta \overline{x^{2}}}{T}. \]

We find:

\[ \begin{array}{lll} \text{for } \Delta = 0.142\ \text{dyne/cm} & k = 1.53 \cdot 10^{-16}\ \text{erg}\cdot\text{degree}^{-1},\\ \text{” } \Delta = 0.0139\ \text{”} & k = 1.32 \cdot 10^{-16}\ \text{”}\ \text{”},\\ \text{” } \Delta = 0.0356\ \text{”} & k = 0.89 \cdot 10^{-16}\ \text{”}\ \text{”},\\ \hline \text{mean (unweighted)} & k = 1.25 \cdot 10^{-16}\ \text{erg}\cdot\text{degree}^{-1}. \end{array} \]

The currently accepted value is \(k = 1.37 \cdot 10^{-16}\ \text{erg}\cdot\text{degree}^{-1}\). The discrepancy is about 8.7%; if greater weight is assigned to the first value (for \(k = 0.142 \dfrac{\text{dyne}}{\text{cm}}\)), obtained from a large number of observations, the discrepancy becomes smaller. In any case, taking into account the nature and conditions of the experiment, it must be admitted that it is already small.

1. The works listed have shown that the spontaneous motion of the moving system of a galvanometer, in magnitude and character, is Brownian motion. To complete the investigation of the question, it was necessary to establish the proportionality of the energy of this motion to temperature, in accordance with the law of equipartition of energy among the degrees of freedom. By this the connection between the arbitrary oscillations of galvanometers and thermal motion would be definitively established.

Unfortunately, direct verification in the simplest (in principle) form—raising or lowering the temperature of the entire circuit, including the temperature of the galvanometer, by several times and determining the mean value of the fluctuations at the new temperature—presents a number of technical difficulties. Therefore Ornstein, Burger, Taylor, and Clarkson[^14] proceeded in a somewhat different way. They connected in series with the galvanometer a coil with coefficient of self-induction \(L_2\) and resistance \(r_2\), and varied its temperature while keeping the temperature of the galvanometer unchanged. Then, as the theory given by Ornstein[^13][^14] shows, the energy of the current fluctuations in the circuit is expressed by the formula:

\[ \frac{(L_1+L_2)i^2}{2} = \frac{k}{2} \left( \frac{r_1}{r_1+r_2}T_1+ \frac{r_2}{r_2+r_2}T_2 \right), \tag{18} \]

where \(L_1, r_1\), and \(T_1\) are the self-induction, resistance, and temperature of the galvanometer, and \(L_2, r_2\), and \(T_2\) are the self-induction, resistance, and temperature of the coil (for the derivation of this formula, see Appendix 1).

For \(T_1=T_2\), formula (18) passes into the form corresponding to the elementary law of uniform distribution of energy. For the potential energy of the galvanometer coil we find, under the assumption that the air resistance is small in comparison with the electromagnetic damping, the very same quantity:

\[ \frac{A\vartheta^2}{2} = \frac{k}{2} \frac{r_1T_1+r_2T_2}{r_1+r_2}. \tag{18′} \]

The latter formula was also subjected to experimental verification. The apparatus consisted of a Moll galvanometer with resistance \(r_1=45\,\Omega\) and a coil of manganin wire of resistance \(137\,\Omega\) or \(400\,\Omega\), wound bifilarly. The motions of the galvanometer were amplified with the aid of a Moll and Burger thermorelay. To eliminate extraneous influences on the circuit, a whole series of various precautions was taken. Thus, the first galvanometer was placed in a special heavy metal box; for protection against electrostatic effects all the wires were enclosed in a lead sheath. The connection of the galvanometer with the manganin coil was made by means of two massive metal bars, insulated from one another only by a thin sheet of mica and wrapped on the outside with cotton wool; this achieved almost complete equality of the temperatures of both contacts and, to a considerable extent, eliminated harmful thermoelectromotive forces. In addition to these and a number of other measures aimed at protecting the primary circuit from disturbances, the causes producing instability of the current in the thermorelay circuit—for example, instability of the radiation of the illuminating lamp—were investigated and to a considerable extent eliminated. As a result, the stability of the apparatus was brought to such a degree that it proved possible to register and quantitatively compare between themselves—

Brownian motion of the galvanometer suspension in two cases: a) a manganin coil at room temperature, b) a coil at the temperature of liquid air. The general scheme of the apparatus is shown in Fig. 4. The curves recorded by the registering apparatus are reproduced in the following Fig. 5. The first four curves confirm the already known influence of the degree of damping on the form of the curve. Curves 5–8 show the influence of the temperature of one section of the circuit on the amplitude of the Brownian motion. In view of the presence of a smooth displacement (“drift”) of the zero point, the authors, in processing these curves, calculated not the deviations \(x\) themselves, but the difference of successive values \(x_1-x_2\).

Fig. 4. Diagram of the connection of the thermorelay in the experiment of Ornstein and collaborators: \(G_1\)—first galvanometer, \(G_2\)—second galvanometer, \(C\)—manganin coil, \(b\)—contact blocks, \(I\)—light source, \(II\) spherical lenses, \(LIII\), \(LIV\)—cylindrical lenses, \(T\)—thermoelement, \(D\)—recording drum, \(P\)—metal box.

Fig. 5. 1,2—galvanometer open; 3,4—galvanometer closed; 5,6—coil in liquid air; 7,8—coil at room temperature.

The mean value of this quantity \(\left|\overline{x_1-x_2}\right|\) differs from \(\overline{|x|}\) only by a numerical factor. To determine \(\left|\overline{x_1-x_2}\right|\), a zero line is drawn approximately by eye, divided into equal intervals, and at the end of these intervals the deviation \(x\) is determined; in forming the differences of successive quantities \(x\), the errors due to the slow displacement of the zero are almost completely eliminated. The values of \(\left|\overline{x_1-x_2}\right|\) thus found from curves 5–6 and 7–8 are related to one another as \(1:0.77\). This ratio agrees satisfactorily with the value given by formula \((18')\), if one substitutes in it: \(r_1=45\,\Omega\), \(T_1=290^\circ\), \(r_2=135\,\Omega\), and \(T_2\)—in one case equal to \(290^\circ\), in the other equal to \(90^\circ\) (liquid oxygen); the calculated ratio is \(1:0.70\). The authors do not determine the absolute values of the deviations or of the current fluctuations corresponding to them, since the apparatus they used with the Moll–Burger thermorelay proved—

yields smaller values than those existing in reality; the reason for this is the thermal and mechanical inertia of the secondary system. But these distortions do not affect the ratio of the mean oscillations at two different temperatures. As for all kinds of jolts, shocks, etc., their presence affects the ratio mentioned, namely, brings it closer to unity (these perturbations do not depend on the temperature of the parts of the circuit, and consequently in both cases one and the same quantity is added to the Brownian motions of the coil). Therefore the experimentally found quantity \(\frac{1}{0.77}\) is closer to unity than the theoretical \(\frac{1}{0.70}\). Taking this into account, we must state that the work of Ornstein and collaborators has given a sufficiently experimental confirmation of the law \(\frac{1}{2}kT\).

9. Establishing the limit of sensitivity of electrical measuring instruments dependent on thermal motion poses for science the problem of how far this limit can be lowered. Formulae (17) and (17′) indicate the steps that must first of all be taken. First, the period of the galvanometer must be made as large as possible. Second, the resistance of the galvanometer must be chosen accordingly: if a high current sensitivity is desired, then \(R\) must be large; if a high voltage sensitivity is needed, then \(R\), on the contrary, must be decreased. However, in practice both parameters can be varied only within certain limits. Excessively large \(\vartheta_0\) are inconvenient for work; very large \(R\) are disadvantageous, since they reduce the deflection attainable for a given type of galvanometer; very small \(R\) are difficult to realize in practice if the requirement of high sensitivity is maintained. The question arises: for given \(\vartheta_0\) and \(R\), what other factors can be used to improve the limiting sensitivity? One of the factors whose significance must be considered is the damping of the galvanometer. In setting forth above the theory of fluctuations of the galvanometer, we assumed that the damping is critical and that air resistance may be neglected in comparison with electromagnetic braking. In one of his subsequent works\(^{11}\) Ising considers the limiting sensitivity of a galvanometer under arbitrary damping. In this case, instead of the equality

\[ \frac{B^{2}}{2KR}=\omega_{0} \]

one must write the more general expression

\[ \frac{B^{2}}{2KR}=n_{2}\omega_{0}, \]

where \(n_{2}\) is a measure of the electromagnetic damping, and for the case of critical damping and in the absence of air ...

of damping \(n_2=1\). The quantity \(n_2\) will enter all subsequent formulae; as a result we shall find:

\[ i_{\min}=4\sqrt{\frac{\pi kT}{R\vartheta_0}\cdot\frac{1}{n_2}} \tag{17''} \]

and analogously

\[ v_{\min}=4\sqrt{\frac{\pi kTR}{\vartheta_0}\cdot\frac{1}{n_2}} . \tag{17'''} \]

Ising raises the question of the lowest threshold of sensitivity not for constant \(\vartheta_0\), but for a constant time of observation \(\tau\). He defines the latter quantity as the interval of time by the end of which the deflection reaches its final value with some preassigned approximation. The quantities \(\tau\) and \(\vartheta_0\) are proportional to one another; the factor of proportionality

\[ \frac{\tau}{\vartheta_0}=f(n) \]

depends only on the quantities

\[ n=\frac{\lambda}{\omega_0} \]

of the measure of complete damping (\(n\) is defined in the same units as \(n_2\), i.e. for the critical case \(n=1\)). Further, \(n\) and \(n_2\) are related by the condition:

\[ n=n_1+n_2; \]

\(n_1\) is a quantity indicating the role of air damping and not dependent on the circuit. The results found by Ising are as follows: if \(n_1\ll 1\), then the sensitivity threshold of the galvanometer for a given \(\tau\) is lowest when the complete damping is close to critical, i.e. \(n\sim 1\). If \(n>1\), then the threshold increases only slightly and even at \(n=\infty\) is only 20% higher than the minimum value. Conversely, when \(n<1\) is decreased, \(i_{\min}\) and \(v_{\min}\) increase very rapidly. For the case of \(n_1\) not very small, one has to take \(n_2>1\), and consequently also \(n>1\), so that \(n\gg n_1\). As a final conclusion, it must be stated that by damping the sensitivity threshold can be lowered only by increasing the time of observation.

M. Czerny\(^{15}\) gave a new, very interesting approach to the question under consideration, which made it possible to illuminate from one very simple point of view all the results obtained and revealed a new essential circumstance. Czerny considers the process of measuring current with a galvanometer first of all as the transformation of part of the electrical energy into the potential energy of a twisted thread or spring. The latter quantity undergoes fluctuations, with the mean (in the absence of a constant current):

\[ \overline{E}_{pot}=\frac{1}{2}\Delta\cdot\overline{\varphi^{2}}=\frac{1}{2}kT . \]

Therefore thermal motion creates an absolute (at a given temperature) limit for the change in the electrical energy supplied to the galvanometer per one division. The latter is equal to

\[ U=VI\tau . \]

Hence we see that it is possible to measure an arbitrarily weak current if the time \(\tau\) and the voltage \(V\) are made sufficiently large; the corresponding assertion also applies to the measurement of \(V\). But here a new circumstance arises. Namely, not all the energy \(U\) supplied to the galvanometer is converted into the potential energy of the coil \(E_{pot}\); in practice always

\[ U>E_{pot}, \]

or one may also write

\[ E_{pot}=\eta\cdot U, \]

where \(\eta<1\); we shall call the quantity \(\eta\) the efficiency of the galvanometer. Since \(E_{pot}\) must be greater than \(\frac{1}{2}kT\) (to satisfy Ising’s requirement: \(x_{\min}=4x_{\text{aver}}\), it is necessary that \(E_{pot}=16\cdot\frac{1}{2}kT\)), which in general is written as

\[ E_{pot}=a\cdot\frac{1}{2}kT \]

(\(a\)—“safety factor”), then

\[ U=\frac{1}{\eta}E_{pot}=\frac{1}{\eta}\cdot a\cdot\frac{1}{2}kT . \tag{18} \]

Czerny points to the necessity, in order to improve the limiting capabilities of electrical measuring instruments, of increasing the efficiency \(\eta\). To clarify this he calculates how much energy is consumed by the galvanometer from the source, how much of it is spent on the work of deflection, and how much on Joule heat. The calculation is carried out for different degrees of damping. The results are so instructive that we shall present part of them, omitting the derivation, which is, however, very simple (consisting in the solution of the equation of motion of the coil).

In the following tables the following are given: \(t\)—the time from the beginning of the reading, \(\frac{\varphi}{\varphi_{\infty}}\)—the ratio of the deflection reached at the moment \(t\) to the final deflection, \(\eta=\frac{E_{pot}}{V^{2}t/r}\)—the ratio of the potential energy to the total energy that could have been obtained from the source by the moment \(t\)*, and finally, \(\frac{Q}{V^{2}t/r}\), where \(Q\) is the Joule heat released. In all cases \(\theta_{0}=40\) sec.

* In reality the energy consumed is somewhat less than \(\frac{V^{2}t}{r}\); this occurs because the counter-electromotive force of induction that develops during motion somewhat slows its expenditure.

These tables show that the least favorable case is that of weak damping. However, even in the best case of critical damping, the greatest value attained by the efficiency coefficient does not exceed 22%; this occurs half a period after the beginning of the measurement, when the deflection has reached only 82% of the final value. According to Cerni’s apt remark, the “greatest action” of a galvanometer consists in heating the room! As Cerni’s “consolation” shows, there exist measuring instruments in which the perceived energy is utilized still much worse. Thus, for example, radiation thermoelements of the most advanced designs, when measuring very weak radiations, cannot give an efficiency coefficient greater than \(10^{-9}\), and in practice \(\eta\) reaches in them values of \(10^{-10}\).

The principal result of Cerni’s work is that any further lowering of the sensitivity threshold of measuring instruments can be expected only from increasing their energy efficiency.

Thermal motion as a source of noise in tube amplifiers.

The remarkable properties of the vacuum tube as an amplifier—

TABLE 4
Case of critical damping \((\lambda=\omega_0,\; n=n_2=1)\)

\(t\) (sec)	\(\dfrac{\varphi}{\varphi_\infty}\)	\(\eta=\dfrac{E_{\text{pot}}}{\dfrac{V^2 t}{r}}\)	\(\dfrac{Q}{\dfrac{V^2 t}{r}}\)
2	0,040	0,005	0,565
6	0,243	0,063	0,274
10	0,466	0,138	[[unclear: value appears as 0,?00]]
16	0,715	0,204	[[unclear: value appears as 0,2?0]]
20	[[unclear: value appears as 0,?21]]	0,214	[[unclear: value appears as 0,?57]]
24	0,890	0,210	0,316
32	0,960	0,184	0,434
40	0,986	0,155	0,531
48	0,995	0,131	0,605
\(\downarrow\)	\(\downarrow\)	\(\downarrow\)	\(\downarrow\)
\(\infty\)	1,000	0,000	1,000

Strongly damped periodic motion \((n_2=10)\)

\(t\) (sec)	\(\dfrac{\varphi}{\varphi_\infty}\)	\(\eta\)
10	0,073	0,034
20	0,146	0,068
30	0,210	0,094
40	0,269	0,115
50	0,325	0,134
100	0,543	0,187
200	0,792	0,199
300	0,906	0,174
400	0,959	0,146
600	0,991	0,104

Weakly damped periodic motion \((n_2=0,1)\)

\(t\) (sec)	\(\dfrac{\varphi}{\varphi_\infty}\)	\(\eta\)
5,7	0,349	0,04
10,7	1,000	0,057
15,7	1,555	0,098
20,7	1,715	0,091
25,8	1,473	0,054
30,8	1,0[[unclear: digit]]	0,021
35,8	0,598	0,006
\(\downarrow\)	\(\downarrow\)	\(\downarrow\)
282	1,000	0,002
297	1,007	0,002
302	1,000	0,002

thereby opened, as is known, a new era in the technology of weak electric currents. The successive use of several tubes makes it possible to achieve amplification of current or voltage by millions of times and more. In the first period after the emergence of this new low-current technology, one might have thought that the power of the cathode tube in the sense of amplifying weak currents was almost unlimited; the question came down only to increasing the number of tubes and the amplification coefficient of each of them. Practice, however, showed that the limit to the perception of extremely weak currents is set by the so-called “noises,” which are always audible if a telephone is connected to the output of such an amplifier. The causes of these noises may be: mechanical and electromagnetic effects on the amplifier, instability of the voltage of the batteries feeding it, nonuniformity of electron emission, imperfection of the insulation in the tubes, the presence of gas residues in them, etc. It is possible to combat these phenomena and to achieve, if not their complete elimination, then in any case a considerable reduction of them.

In 1918 Schottky⁶ pointed out that even with the complete elimination of all the causes listed above, a certain noise background in the amplifier must nevertheless remain. Its causes are fundamentally irremovable and lie in the very nature of things. These are, first, the thermal motion of charges in conductors already described above, and, second, the inhomogeneity of the electron current in the tube, depending on the atomic nature of electricity. The latter phenomenon, to which Schottky gave the name “shot effect” (Schrot-effekt), will be the subject of the second part of the present survey. For now we shall consider the influence of thermal motion.

Let some conductor be connected between the grid and the cathode of the first tube of the amplifier. Such a conductor may be, for example, a circuit connected with the antenna of a receiving radio station. In this conductor, as a consequence of thermal motion, electromotive forces arise which depend on its temperature and resistance; see formula (9). At its ends, connected to the tube, there appears the corresponding potential difference, causing a change in the electron current in the tube. These changes of current, subsequently amplified by the following tubes, create the noise background. If the antenna receives a signal which on the grid of the first tube gives a voltage smaller than that which arises owing to fluctuations, then the signal will be drowned in the noise background and cannot be discerned. Schottky estimates the minimum power required at the input of the amplifier in order to obtain an audible signal in the following way. The energy of the alternating current which must be present in the input circuit in order to give a clearly audible tone at the output must be no less than the fluctuation energy, i.e.,

\[ \frac{Li^2}{2}=\frac{kT}{2}. \]

The power required to maintain such a current is:

\[ W=\overline{r i^2}=\frac{r}{L}kT, \tag{20} \]

where \(\frac{r}{L}\)—the damping constant of the circuit—in practice has a value of the order of \(10^3\)—\(10^4\ \mathrm{sec}^{-1}\). Taking it equal to \(10^3\ \mathrm{sec}^{-1}\) and \(kT=4\cdot10^{-14}\ \mathrm{erg}\), we find:

\[ W=4\cdot10^{-11}\frac{\mathrm{erg}}{\mathrm{sec}}=4\cdot10^{-18}\ \mathrm{W}.* \]

In practice the threshold of power accessible to reception and reproduction usually lies higher, for the reasons already mentioned.

The thermal effect in an amplifier in almost pure form was observed and studied by J. B. Johnson\(^{18–20}\). He succeeded in showing that part of the noise produced by an amplifier depends on the input resistance, and not on the tubes. Proceeding from the fact that this component of the noise, as is known from practice and follows from theory, increases together with the resistance, Johnson used in his work input resistances of \(100\,000\ \Omega\) and higher. Under these conditions the thermal motion considerably exceeded the other sources of noise.

Fig. 6. Schematic diagram of Johnson’s experiments.

The schematic arrangement with which Johnson worked is very simple (Fig. 6). A high-ohmic resistance was connected to the grid and cathode of the first tube of the amplifier. At the output of the amplifier, by means of a transformer, a vacuum thermoelement connected to a direct-current galvanometer was included. The current supplied by the thermoelement was exactly proportional to the square of the current that passed through it; therefore the galvanometer readings were determined by the square of the voltage at the input of the amplifier. This circumstance is very important, since it makes it possible to observe experimentally a quantity introduced theoretically.

Unfortunately, Johnson gives very little information about the details of his apparatus. The amplifier was a six-stage one, with different tubes in the different stages. The transitions between them were through transformers or chokes. The only exception was one stage containing either a tuned circuit, connected as shown in Fig. 7, or a band-pass filter passing frequencies from 500 to \(1000\ \mathrm{sec}^{-1}\). The entire amplifier as a whole was carefully shielded from mechanical, acoustic, and electromagnetic external influences. The frequency characte-

* We shall see below that a more detailed theory of Nyquist leads to a somewhat different expression.

the amplifier’s characteristic could be regulated both in width and in the position of the maximum amplification by varying one of the elements of the resonant circuit \((L_1, L_2, R)\). The calibration was carried out over a limited frequency interval (for example, from 200 to 2200 sec.\(^{-1}\)), but one sufficiently large for the remaining frequencies to contribute only an insignificant fraction of the whole area covered by the frequency characteristic.

The theory developed by Nyquist\(^{21,22}\) shows that the current at the output of the amplifier is expressed by the following equality:

\[ \overline{J^{2}}=\frac{2kT}{\pi}\int_{0}^{\infty} R(\omega)Y^{2}(\omega)\,d\omega, \tag{21} \]

where \(R(\omega)\) is the watt component of the input resistance, \(Y(\omega)\) is the ratio of the current at the output to the voltage causing it at the input at frequency \(\omega\); \(Y=\dfrac{di_{\text{out}}}{dv_{\text{in}}}\), in other words, the slope of the amplitude characteristic of the amplifier. Nyquist’s derivation, in our opinion, cannot be considered indisputable (see Appendix 2). However, the formula obtained by him was well confirmed by Johnson’s experiments.

Fig. 7. Johnson’s tuned amplifier cascade.

As the resistances under test, samples of various materials were tried: carbon filaments, wires of Advens*, platinum and copper layers deposited by evaporation on glass, then commercial grid-leak resistors, and finally electrolytic resistances—solutions of NaCl, CuSO\(_4\), K\(_2\)Cr\(_2\)O\(_4\), Ca(NO\(_3\))\(_2\) in water and H\(_2\)SO\(_4\) in ethyl alcohol. The magnitudes of the resistances varied from \(0.5\cdot 10^{5}\) to \(9\cdot 10^{5}\ \Omega\). The full value of the watt resistance is determined, in addition to the ohmic resistance \(R_0\), also by the fact that a capacitance \(C\), composed of the capacitance of the conductors and the input capacitance of the amplifier, is connected in parallel with it in the input circuit of the amplifier. In this case, as is easy to calculate,

\[ R(\omega)=\frac{R_0}{1+\omega^{2}C^{2}R_0^{2}}. \]

The quantities \(R_0\) and \(C\) in each experiment were measured directly in place, i.e., after connecting the resistance to the amplifier.

* American Advens, very close in composition and properties to constantan.

The definite integral appearing in equality (21) was calculated graphically. In doing so, a correction was introduced for that part of the area which had not been measured experimentally. In cases of sharp tuning of the amplifier, when the integrand differs strongly from zero only over a small frequency interval, \(R(\omega)\) may be taken outside the integral sign and its value taken for some mean \(\omega\); then graphically it remains to calculate only

\[ \int_0^\infty Y^2(\omega)\,d\omega . \]

If such sharp tuning is kept unchanged, then, by varying \(R_0\), one can study the dependence of \(\overline{J^2}\) on \(R\). In this case there is not even any need to know the entire frequency characteristic; it is sufficient to control only one point, for example the maximum of amplification. Knowing \(Y_{\max}\), one can, by measuring \(\overline{J^2}\) at the output, calculate such a \(\overline{V^2}\) at the input as would have produced the given \(\overline{J^2}\), and regard this quantity \(\overline{V^2}\) as a measure of the fluctuations in the input circuit. The results are presented in Figs. 8 and 9.

• ● Carbon filament
• • Wire of alpaca
• × \(\mathrm{CuSO_4}\) (solution)
• ⊙ \(\mathrm{K_2Cr_2}\), ”
• ○ \(\mathrm{Ca(NO_3)}\), ”

Fig. 8. Dependence of the thermal effect on the active component of the input resistance.

Fig. 9. Dependence of the thermal effect on the ohmic resistance of the input circuit.

The first of them represents the dependence of \(\overline{V^2}\) on the active resistance \(R(\omega)\) for various substances. We see from this figure, first, that the magnitude of the fluctuations at a given resistance does not depend on the substance of the conductor, nor even on the kind of conductivity—ionic or electronic; second, that the mean square of the voltage in the input circuit is, on the average, very accurately proportional to the resistance. The quantity

\[ \overline{W}=\frac{\overline{V^2}}{R} \]

—the equivalent power at the input—turns out to be constant in all these cases. The second figure (9) represents the dependence of \(\overline{V^2}\) on \(R_0\) for a given \(C\); the experimental points lie close to the calculated curve.

Next, Johnson determined the dependence of \(\dfrac{\overline{V^2}}{R_0}\) on the absolute temperature. For this purpose the temperature of the resistance was varied from \(-18^\circ\) C (liquid air) to \(100^\circ\) C (boiling water). Figure 10 gives the results for three wires with different resistances, made of advance.

In all three cases the equivalent fluctuating power proved to be one and the same linear function of the temperature. The same result was obtained with all the other tested resistances, both solid and liquid (electrolytes). This fact is the most convincing proof of the thermal origin of the phenomenon and of the applicability to it of the law of equipartition of energy over the degrees of freedom. We note that the accuracy of the results in these experiments is considerably higher than in the work with galvanometers.

\[ \frac{\overline{V^2}}{R(\omega)} \;(\text{watt}) \]

Legend:

\[ \square\quad R(\omega)=0.15\cdot 10^{-6}\ \Omega \]

\[ \circ\quad R(\omega)=0.29\cdot 10^{-6}\ \Omega \]

\[ +\quad R(\omega)=0.44\cdot 10^{-6}\ \Omega \]

Fig. 10. Dependence of the thermal effect on temperature.

To verify definitively the accuracy with which equation (21) is confirmed, Johnson carried out a series of absolute measurements with an accurate determination of the area of the frequency characteristic, and from them calculated, using equation (21), the value of Boltzmann’s constant \(k\). The results are given in Table 5.

The mean of the values found is

\[ k=1.27\cdot 10^{-16}\ \frac{\text{erg}}{\text{grad}} \]

with an average error of \(13\%\). The value currently accepted,

\[ k=1.37\cdot 10^{-16}\frac{\text{erg}}{\text{grad}}, \]

differs by \(7.5\%\) from that obtained from Johnson’s experiments. Taking into account that all values of \(k\) obtained at various frequencies from 295 to \(1830\ \text{sec}^{-1}\) prove to be rather close to the true value, formula (21) may be considered experimentally verified, and together with it the very conception of the essence of the phenomenon.

Formula (21) makes it possible to form a more accurate picture of the distribution of the fluctuating electromotive force over the spectrum. Suppose that the input resistance does not depend on the frequency, \(R(\omega)=R=\text{const}\), and further, that the amplifier has

TABLE 5

No.	Frequency (sec.\(^{-1}\))	\(T(^{\circ}K)\)	\(R(\omega)\cdot 10^{-6}\ \Omega\)	\(\displaystyle \int_0^\infty Y^2(\omega)d\omega \times 10^{-10}\)	\(\displaystyle \int_0^\infty R(\omega)Y^2(\omega)d\omega \times 10^{-16}\)	\(\Delta S\) (% \(v_0\))	\(\overline{J^2}\cdot 10^6\) (amp.)	\(k\cdot 10^{16}\) erg/degree
1	1010	298	0,526	0,213		12,0	2,7	1,27
2	2023	″	0,470	0,272		26,2	2,8	1,15
3	1418	″	0,508	0,361		15,6	3,8	1,09
4	″	″	″	0,188		42,3	1,8	0,99
5	295	″	0,548	0,252		3,3	3,1	1,18
6	″	″	″	0,202		15,8	2,0	0,95
7	302	″	″	0,221		12,4	2,7	1,18
8	″	″	″	0,195		15,3	2,3	1,13
9	653	″	0,541	0,747		6,8	10,4	1,14
10	″	″	″	0,645		12,9	8,6	1,30
11	1418	″	0,508	0,236		18,5	3,5	1,26
12	″	″	″	0,161		41,3	1,7	1,09
13	1465	″	0,505	1,93		18,7	21,2	1,14
14	″	″	″	1,75		20,3	19,1	1,14
15	635	″	0,541	0,594		7,8	8,9	1,46
16	″	″	″	0,139		35,0	2,1	1,47
17	″	″	″	0,597		10,9	7,8	1,28
18	643	295	0,44		0,439	0	11,0	1,38
19	645	297	″		0,396	0	11,1	1,49
20	1830	301	″		0,913	0	19,8	1,13
21	500—1060	299	″		0,331	0	25,0	1,64
22	″	300	″		0,662	0	21,5	1,70
23	″	300	″		0,832	0	26,0	1,63

frequency characteristic of the following form: \(Y(\omega)=0\) for \(0<\omega<\omega_1\), \(Y(\omega)=Y=\mathrm{const}\) for \(\omega_1<\omega<\omega_2\), and, finally, \(Y(\omega)=0\) for \(\omega>\omega_2\), i.e., the amplifier passes only the band from \(\omega_1\) to \(\omega_2\), and within it the amplification is constant. Then

\[ \overline{J^2}=\frac{2kT}{\pi}RY^2(\omega_2-\omega_1)=4kTRY^2(f_2-f_1), \tag{21′} \]

where \(f_1\) and \(f_2\) are the frequencies corresponding to the edges of the pass band,

\[ f_1=\frac{\omega_1}{2\pi},\quad f_2=\frac{\omega_2}{2\pi}. \]

Since \(Y\) is the ratio of the output current to the voltage at the input that produces it, it follows from (21′) that the fluctuation voltages acting at the input of the amplifier are equivalent to an alternating voltage with a frequency lying between \(f_1\) and \(f_2\), determined by the following equality:

\[ V^2=4kTR(f_2-f_1). \tag{22} \]

This quantity may be taken as the mean square of the effective electromotive force of the fluctuations in the given frequency interval and written as:

\[ \overline{E^2}=4kTR(f_2-f_1). \tag{22′} \]

From formulas (22′) it is clear that the quantity $\overline{E^2}$ is proportional to the width of the frequency interval $f_2-f_1$ and does not depend on the frequency itself. An analogous circumstance always occurs when decomposing into a spectrum the square of a quantity having a completely random character; we shall encounter the same case in Part II of the present review when considering the shot effect.

Finally, let us determine the equivalent power of the fluctuations for the given frequency interval:

\[ W=\frac{E^2}{R}=4kT(f_2-f_1), \tag{23} \]

i.e. the equivalent power is proportional to the absolute temperature and to the frequency interval, and no longer depends on any other factors. To estimate the order of magnitude of $W$ and $\overline{E^2}$, let us take, together with Johnson: $T=300^\circ \mathrm{K}$, $f_2-f_1=5000\ \mathrm{sec}^{-1}$ (the usual range of audio transmission), $R=0.5\cdot 10^6\Omega$.

Then

\[ W=1.6\cdot 10^{-13}\cdot 5000 =8\cdot 10^{-10}\ \frac{\mathrm{erg}}{\mathrm{sec}} =0.8\cdot 10^{-16}\ \mathrm{W}. \]

\[ \overline{E^2}=W\cdot R=0.4\cdot 10^{-10}\ \mathrm{V}^{-2}, \]

\[ E_{\mathrm{eff}}=\sqrt{\overline{E^2}}\simeq 0.64\cdot 10^{-5}\ \mathrm{V}. \]

Formulas (22′) and (23) indicate the measures that can be used to reduce noise from thermal motion. These are: 1) to reduce the input resistance, 2) to reduce its temperature, 3) to narrow the passband of the amplifier to the absolutely necessary limits. The greatest practical interest is presented by the last indication. In those cases where the amplification of a current of a definite frequency is involved, it is advantageous to use an amplifier with extremely sharp tuning.

At the end of his work Johnson points out that the source of thermal noise may be not only the external resistance connected to the grid of the tube, but also the internal resistance of the tube itself. If the resistance in the grid circuit is sufficiently large, then this new source of noise plays a secondary role; however, when the external resistance is reduced, its significance increases. This question is considered in more detail in the work of Llewellyn[^23].

Summarizing this part of the review, one may state that the existence of electrical fluctuations as a consequence of thermal motion has been established experimentally by various methods, and that their magnitude has proved to correspond to the conclusions of the theory based on the law of the uniform distribution of energy over degrees of freedom. To a considerable extent their role has been clarified as a limit to the action of electrical instruments, and those possibilities have been analyzed which must be used for further pushing back of this limit.

Addendum 1

L. S. Ornstein \(^{13,14}\) derived a formula for the mean energy of fluctuations in a circuit containing sections at different temperatures, using the new method he had found for considering fluctuation problems. Let us first give Ornstein’s derivation for the case of a uniformly heated circuit, in order to acquaint the reader with this method*, and then—for the case of different temperatures.

1. We write the equation of the current in a circuit containing self-inductance and resistance, as above:

\[ L\frac{di}{dt}+ri=E(t), \tag{A} \]

where \(E\) is the fluctuating electromotive force. Integrating this equation:

\[ i=i_0 e^{-\frac{r}{L}t}+\frac{1}{L}e^{-\frac{r}{L}t}\int_0^t E(\xi)e^{\frac{r}{L}\xi}\,d\xi . \tag{B} \]

Let us compute the mean square of the current at the time \(t\), taking the initial current \(i_0\) as given:

\[ \overline{i^2}=i_0^2 e^{-\frac{2r}{L}t} +\frac{1}{L^2}e^{-\frac{2r}{L}t} \overline{\left\{\int_0^t E(\xi)e^{\frac{r}{L}\xi}\,d\xi\right\}^2}. \tag{C} \]

The term containing the integral to the first power vanishes, since

\[ \overline{E(\xi)}=0. \]

The mean square of this integral can be transformed as follows:

\[ \overline{\left\{\int_0^t E(\xi)e^{\frac{r}{L}\xi}\,d\xi\right\}^2} = \int_0^t\int_0^t \overline{E(\xi)\cdot E(\eta)} e^{\frac{r}{L}(\xi+\eta)}\,d\xi\,d\eta . \]

The mean value \(\overline{E(\xi)\cdot E(\eta)}\) is different from zero, but only for very close instants of time \(\xi\) and \(\eta\), since, owing to the complete arbitrariness of the function \(E(t)\), for more distant instants the values of this function are independent of one another. Therefore one may put

\[ \eta=\xi+\varphi, \]

* As far as we know, it has not yet been expounded in Russian.

where \(\psi\) is a very small quantity.

\[ \int_{0}^{t}\int_{0}^{t}\overline{E(\xi)\cdot E(\eta)}\cdot e^{-\frac{r}{L}(\xi+\eta)}\,d\xi\,d\eta = \int \overline{E(\xi)\cdot E(\xi+\psi)}\,d\psi \int_{0}^{t} e^{\frac{2r}{L}\xi}\,d\xi . \]

The first integral on the right-hand side does not depend on \(\xi\); moreover, the integrand becomes zero for all \(\psi\) that are not very small. We may therefore take the limits of integration from \(-\infty\) to \(+\infty\) and introduce the notation

\[ \int_{-\infty}^{+\infty} E(\xi)\cdot E(\xi+\psi)\,d\psi=\overline{EE}. \tag{D} \]

Further,

\[ \int_{0}^{t} e^{\frac{2r}{L}\xi}\,d\xi = \frac{L}{2r}\left(e^{\frac{2r}{L}t}-1\right). \]

Therefore equation (C) is rewritten in the following form:

\[ i^{2}=i_{0}^{2}\cdot e^{\frac{2r}{L}t} + \frac{1}{2rL}\left(1-e^{\frac{2r}{L}t}\right)\overline{EE}. \tag{E} \]

Multiplying this expression by \(\frac{L}{2}\), we obtain the mean energy of the fluctuation currents, which, by the law of equipartition of energy, must be equal to \(\frac{kT}{2}\). Let a sufficiently long interval of time have elapsed from the beginning of the process: \(t\to\infty\). Then from (E) we find:

\[ \frac{L\overline{i^{2}}}{2} = \frac{1}{2rL}\overline{EE}\cdot\frac{L}{2} = \frac{1}{4r}\overline{EE} = \frac{kT}{2}, \]

whence

\[ \overline{EE}=2rkT. \tag{F} \]

Using exactly the same method, Ornstein proves that for a mechanical system undergoing Brownian motion under the action of a random force \(F(t)\), which represents molecular impacts, one may write:

\[ \overline{FF} = \int_{-\infty}^{+\infty} F(\xi)\cdot F(\xi+\psi)\,d\psi = 2\beta kT, \tag{F'} \]

where \(\beta\) is the coefficient of friction experienced by the given system.

1. Next, let us consider the motion of the galvanometer coil with a closed circuit, which is in Brownian motion for two reasons: current fluctuations and impacts of air molecules. The equations of motion of this coil are:

\[ K\ddot{\vartheta}+\beta\dot{\vartheta}+A\vartheta+Bi=F, \]

\[ L\frac{di}{dt}+ri-B\dot{\vartheta}=E, \tag{G} \]

where the notation has been chosen in accordance with the preceding: \(\vartheta\) is the angle of deflection of the coil, \(K\) is the moment of inertia of the coil, \(A\) is the directing force, \(B\) is the dynamic constant, \(\beta\) is the coefficient of friction; the remaining notation is clear. As has just been derived:

\[ \overline{FF}=2\beta kT,\quad \overline{EE}=2rkT; \]

the connection between the mechanical and electrical parameters of the system does not affect these quantities. Further, owing to the complete independence of the fluctuation impulses of the one and the other nature:

\[ \overline{EF}=0. \tag{F''} \]

Restricting himself to the case of large \(r\) and critical damping: \(4KA=\left(\beta+\dfrac{B^{2}}{r}\right)^2\), Ornstein shows that for \(\dfrac{A\vartheta^{2}}{2}\), \(\dfrac{K\dot{\vartheta}^{2}}{2}\), and \(\dfrac{Li^{2}}{2}\) one obtains values equal to \(\dfrac{kT}{2}\). To this end he first solves equation (G) without right-hand sides by the substitution \(\vartheta=ae^{pt}\), \(i=be^{pt}\), and for \(p\) three values are found and a relation between \(a\) and \(b\) is established. Next, by the method of variation of constants, the values \(a_1,a_2,a_3\) are determined so as to satisfy equations (G) with the right-hand sides. It turns out that, for large \(t\),

\[ i=\frac{1}{L}e^{-\frac{r}{L}t}\int_0^t e^{\frac{r}{L}\xi}E(\xi)\,d\xi \]

and

\[ \vartheta=\frac{1}{p_1-p_2}\frac{1}{K}\left\{ e^{-p_1t}\int_0^t e^{p_1\xi}\left[F(\xi)-\frac{B}{r}E(\xi)\right]d\xi - e^{-p_2t}\int_0^t e^{p_2\xi}\left[F(\xi)-\frac{B}{r}E(\xi)\right]d\xi \right\}, \]

where

\[ p_1=-\omega+\frac{BL}{r},\quad p_2=-\omega-\frac{BL}{r},\quad \text{and}\quad \omega=\sqrt{\frac{A}{K}} \]

is the natural frequency of the galvanometer coil. Let us note that the magnitude of the current \(i\) in this case coincides exactly with expression (B)

for sufficiently large \(t\), when the first term may be neglected. This coincidence occurred because \(r\) was taken to be very large; it is natural that, in this case, the electromotive forces induced owing to the rotation of the coil cannot produce appreciable currents, whereas the electromotive forces \(E\) arising in the conductor of the coil itself as a result of thermal motion increase together with \(r\). Therefore here, as in the preceding case:

\[ \overline{i^2}=\frac{1}{L}kT \]

and

\[ \frac{L\overline{i^2}}{2}=\frac{kT}{2}. \]

Further,

\[ \vartheta^2=\frac{1}{(p_1-p_2)^2}\frac{1}{K^2} \left\{ e^{-2p_1t}\left[ \int_0^t e^{p_1\xi}\left(F-\frac{B}{r}E\right)d\xi \right]^2 +\right. \]

\[ \left. +e^{-2p_2t}\left[ \int_0^t e^{p_2\xi}\left(F-\frac{B}{r}E\right)d\xi \right]^2 -2e^{-(p_1+p_2)t} \int_0^t e^{p_1\xi}\left(F-\frac{B}{r}E\right)d\xi \times \int_0^t e^{p_2\xi}\left(F-\frac{B}{r}E\right)d\xi \right\} = \]

\[ =\frac{1}{(p_1-p_2)^2}\frac{1}{K^2} \left\{ e^{-2p_1t}\cdot\frac{1}{2p_1}(e^{2p_1t}-1) \left(\overline{FF}+\frac{B^2}{r^2}\overline{EE}\right) +\right. \]

\[ \left. +e^{-2p_2t}\frac{1}{2p_2}(e^{2p_2t}-1) \left(\overline{FF}+\frac{B^2}{r^2}\overline{EE}\right) - 2e^{-(p_1+p_2)t}\frac{1}{p_1+p_2} \left(e^{(p_1+p_2)t}-1\right) \left(\overline{FF}+\frac{B^2}{r^2}\overline{EE}\right) \right\}. \]

Carrying out the simplifications, substituting their values for \(p_1\) and \(p_2\), replacing \(\overline{FF}=2\beta kT\) and \(\overline{EE}=2rkT\), and putting \(t=\infty\), we find:

\[ \overline{\vartheta^2} = \frac{1}{(p_1-p_2)^2}\frac{1}{K^2} \left\{ \frac{1}{2p_1}+\frac{1}{2p_2}-\frac{2}{p_1+p_2} \right\} \left(2\beta kT+\frac{B^2}{r^2}2rkT\right) = \]

\[ = \frac{kT}{K^2p_1p_2(p_1+p_2)} \cdot \left(\beta+\frac{B^2}{r}\right) = \frac{kT}{2\omega\left(\omega^2-\frac{B^2L^2}{r^2}\right)K^2} \left(\beta+\frac{B^2}{r}\right). \]

Taking into account that, according to our assumptions,

\[ \frac{B^2L^2}{r^2}\ll \omega^2 \]

and

\[ \beta+\frac{B^2}{r}=2\sqrt{AK}=2K\omega, \]

we obtain:

\[ \overline{\vartheta^2}=\frac{kT}{K\omega^2}=\frac{kT}{A}, \]

whence, finally,

\[ \frac{A\theta^2}{2}=\frac{kT}{2}. \]

1. We pass to the case of a circuit containing two sections with different temperatures. We denote the resistances of the sections by \(r_1\) and \(r_2\), the coefficients of self-induction by \(L_1\) and \(L_2\), the temperatures by \(T_1\) and \(T_2\), and the fluctuation electromotive forces by \(E_1\) and \(E_2\):

\[ (L_1+L_2)\frac{di}{dt}+(r_1+r_2)i=E_1+E_2. \tag{H} \]

Further, we have:

\[ \begin{aligned} \overline{E_1E_1}&=2r_1kT,\\ \overline{E_2E_2}&=2r_2kT,\\ \overline{E_1E_2}&=0. \end{aligned} \tag{I} \]

Solving (H), we find:

\[ i=i_0 e^{-\frac{r_1+r_2}{L_1+L_2}t} +\frac{1}{L_1+L_2} +e^{-\frac{r_1+r_2}{L_1+L_2}t} \int_0^t e^{\frac{r_1+r_2}{L_1+L_2}\xi}\,[E_1(\xi)+E_2(\xi)]\,d\xi. \]

Let us take the mean square of \(i\), again putting \(t\) sufficiently large so that the term \(i_0\) disappears:

\[ \overline{i^2}= \frac{e^{-2\frac{r_1+r_2}{L_1+L_2}}}{(L_1+L_2)^2} \int_0^t\int_0^t e^{\frac{r_1+r_2}{L_1+L_2}(\xi+\eta)} \times \]

\[ \times \overline{[E_1(\xi)+E_2(\xi)]\,[E_1(\eta)+E_2(\eta)]}\,d\xi\,d\eta = \]

\[ = \frac{1}{2(L_1+L_1)(r_1+r_2)}\cdot(2kr_1T_1+2kr_2T_2). \]

It follows from this that:

\[ \frac{L_1+L_2}{2}\,\overline{i^2} = \frac{k}{2}\,\frac{r_1T_1+r_2T_2}{r_1+r_2}. \]

This is formula (18), given in the text on p. 832.

1. Finally, let us consider a galvanometer with resistance \(r_1\) and temperature \(T_1\) and an external circuit connected to it with resistance \(r_2\) and temperature \(T_2\). We neglect the self-induction of the circuit. The equations of motion of the system are:

\[ \begin{aligned} K\theta+\beta\dot{\theta}+A\ddot{\theta}+Bi&=F,\\ (r_1+r_2)i-B\dot{\theta}&=E_1+E_2. \end{aligned} \tag{J} \]

The quantities \(F_1,\ E_1\) and \(E_2\) are characterized by the relations:

\[ \overline{FF}=2\beta kT_1, \qquad \overline{FE_1}=0, \]

\[ \overline{E_1E_1}=2r_1kT_1, \qquad \overline{E_1E_2}=0, \tag{K} \]

\[ \overline{E_2E_2}=2r_2kT_2, \qquad \overline{FE_2}=0, \]

From equations (6) we eliminate \(i\); we find:

\[ K\ddot{\vartheta} + \left(\beta+\frac{B^2}{r_1+r_2}\right)\dot{\vartheta} + A\vartheta = F-\frac{B}{r_1+r_2}(E_1+E_2); \]

denoting

\[ \frac{\beta}{K}+\frac{B^2}{(r_1+r_2)K}=2\lambda, \qquad \frac{A}{K}=\omega_0^2, \qquad \frac{F}{K}-\frac{B}{K(r_1+r_2)}(E_1+E_2)=G, \]

we rewrite the preceding equation in the form:

\[ \ddot{\vartheta}+2\lambda\dot{\vartheta}+\omega_0^2\vartheta=G. \]

Solving this equation and substituting the values of \(\lambda,\ \omega_0\) and \(G\), we find, still assuming \(t\) to be very large:

\[ \overline{\vartheta^2} = \frac{k}{A\left(\beta+\dfrac{B^2}{r_1+r_2}\right)} \left[ \beta T_1+ \frac{B^2}{(r_1+r_2)^2}(r_1T_1+r_2T_2) \right]. \]

If we neglect the mechanical friction \(\beta\) in comparison with the electromagnetic damping \(\dfrac{B^2}{r_1+r_2}\), then we may write:

\[ \overline{\vartheta^2} = \frac{k}{A\dfrac{B^2}{r_1+r_2}} \cdot \frac{B^2}{(r_1+r_2)^2} (r_1T_1+r_2T_2) = \frac{k(r_1T_1+r_2T_2)}{A(r_1+r_2)}, \]

whence the mean potential energy is

\[ \frac{A\overline{\vartheta^2}}{2} = \frac{k}{2}\, \frac{r_1T_1+r_2T_2}{r_1+r_2}, \]

—the same quantity that we found in the preceding case for the mean energy of the current.

Addendum II.

The formula for the electromotive force of fluctuations as a function of frequency, from which, as a consequence, the formula verified by Johnson is obtained, was derived by H. Nyquist by considering a certain thought experiment.

Nyquist considers two conductors with identical resistance \(R\) and temperature \(T\), connected by long wires having no resistance and emitting no radiation (Fig. 11).

The distributed capacitance \(C\) and self-inductance \(L\) (per unit length) are chosen so that

\[ \sqrt{\frac{L}{C}}=R; \]

ELECTRICAL FLUCTUATIONS AND THE LIMIT OF SENSITIVITY

under these conditions, as is known, no reflection will occur from the ends of the line. The electromotive forces arising in each of the conductors produce in the line two streams of waves traveling in one and the other direction; these electromagnetic waves carry energy from one conductor to the other. It is easy to see that the amount of energy imparted per unit time to conductor II from conductor I is, on the average, equal to the reverse flow of energy from conductor II to I; otherwise we would observe, with time, a change in the temperature of two conductors initially at the same temperature. Further, it is not difficult to establish that not only the entire power of the fluctuations developed by each of the conductors, but also the part of it falling within a given frequency interval, must likewise be the same for both conductors, i.e. \(U(\nu)\,d\nu\) can depend only on the temperature. Indeed, suppose that this is not so and that for some frequency \(\nu\) the inequality \(U_I(\nu) \gtrless U_{II}(\nu)\) holds. In this case the total \(U_I=\int_0^\infty U_I(\nu)\,d\nu=U_{II}\). Then, by including in the line some circuit having no watt resistance and transmitting predominantly the frequency \(\nu\), we would have to obtain a prevailing flow of energy in one particular direction. But, according to what was proved above, this is impossible. Consequently, the function \(U(\nu)\,d\nu\) must be the same for both conductors.

Fig. 11.

Let us imagine that, at some moment of time after the establishment of thermal equilibrium in the whole system, the line is suddenly disconnected from conductors I and II by breaking it or, on the contrary, by short-circuiting it. Then the waves moving along the line will be wholly reflected from the ends, and the energy that was in the line at the moment of disconnection will be preserved in it. After some time, standing waves of all possible frequencies in the line will be established, i.e. \(\frac{\nu}{2l}, \frac{2\nu}{2l}, \frac{3\nu}{2l}\), etc. For each of these frequencies, as a result of thermal equilibrium, there must be an energy \(kT\). In the frequency interval \(d\nu\) there are

\[ d\nu : \frac{\nu}{2l}=\frac{2l\,d\nu}{\nu} \]

proper frequencies of the circuit. Consequently, the total energy of the oscillations in this interval is

\[ \frac{2l\,d\nu}{\nu}\,kT. \]

In the case of an open line, this amount of energy is delivered to both conductors during the time required for the wave to pass along the wires in one direction, i.e. during the time \(t=\frac{l}{\nu}\).

Therefore the power consumed and, consequently, released by each of conductors I and II is

\[ U(\nu)\,d\nu=kT\,d\nu . \tag{L} \]

Thus the fluctuation power is distributed uniformly over all frequencies. On the other hand, one may write:

\[ U(\nu)\,d\nu=\frac{\overline{E^{2}(\nu)\,d\nu}}{4R^{2}(\nu)}\,R(\nu). \]

Consequently,

\[ \overline{E^{2}(\nu)\,d\nu}=4R(\nu)\,U(\nu)\,d\nu=4R(\nu)kT\,d\nu . \]

The same expression for the quantity \(\overline{E^{2}(\nu)\,d\nu}\) must hold in all conductors.

Let a conductor in which such fluctuations occur be connected to the input of an amplifier; let the slope of the amplifier characteristic be \(Y(\nu)\). Then the mean square of the current at the amplifier output in the given frequency interval is:

\[ \overline{I^{2}(\nu)}\,d\nu=\overline{E^{2}(\nu)}\,Y^{2}(\nu)\,d\nu =4R(\nu)kT\,Y^{2}(\nu)\,d\nu \]

or, passing to the variable \(\omega=2\pi\nu\), \(d\omega=2\pi\,d\nu\):

\[ \overline{I^{2}(\nu)}\,d\nu =4R(\omega)\,kT\,Y^{2}\,\frac{d\omega}{2\pi} =\frac{2}{\pi}R(\omega)kT\,Y^{2}(\omega)\,d\omega . \]

Integrating both sides of the equality from \(0\) to \(\infty\), we find:

\[ \overline{I^{2}} =\frac{2}{\pi}kT\int_{0}^{\infty}R(\omega)Y^{2}(\omega)\,d\omega \]

—formula (21), p. 840.

Literature

1. A) A. Einstein. Ann. der Phys., 33, 1910.
   M. Smoluchowski. Ann. der Phys., 25, 205, 1908.
   B) Exposition of the theory of fluctuations—in most textbooks of kinetic theory; in Russian:
   A. K. Timiryazev, Kinetic Theory of Matter, lecture 20;
   E. Bloch, Kinetic Theory of Gases, Ch. VIII. Specially devoted to fluctuations is the book:
   R. Fürth, Schwankungserscheinungen in der Physik. Published by Fr. Vieweg, in the series “Die Wissenschaft.”

2. A. Einstein. Ann. der Phys., 22, 571, 1907.

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4. A. Einstein und Hopf. Ann. der Phys., 33, 1105, 1910.

5. A. Einstein. Ann. der Phys., 14, 360, 1914.

6. W. Schottky. Ann. der Phys., 57, 541, 1918.

7. W. J. Moll and H. C. Burger. Phil. Mag. (7), V. 1, p. 630, 1925.

8. G. Ising, Undersökningar rörande elektrometrar (Swedish), I, p. 46, 1917; II, p. 363, 1919.

9. G. Ising. Kosmos, p. 69, 1923.

10. ” Phil. Mag. 1, p. 827, 1926.

11. ” Ann. der Phys., 8, S. 905 and 911, 1931.

12. F. Zernike. Ztschr. für Phys. 40, S. 628, 1927; 14, 755, 1932.

13. L. S. Ornstein. Ztschr. für Phys., 41, S. 818, 1927.

14. L. S. Ornstein, H. C. Burger, J. Taylor and W. Clarkson. Proc. Roy. Soc., (A). V. 115, p. 391, 1927.

15. M. Czerny. Ann. der Phys., 12, 993, 1932.

16. W. J. H. Moll und H. C. Burger. Ztschr. für Phys., 32, S. 412, 1925.

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1. More precisely, one should speak not of a limit of sensitivity, but of the smallest quantity of electrical energy available for measurement. Brownian motion imposes no restrictions on sensitivity in its usual sense, i.e. on the value \(\dfrac{\partial x}{\partial q}\), where \(x\) is the reading of the instrument and \(q\) is the measured quantity. However, such terminology—“limit of sensitivity”—is current among the majority of authors writing on this question. ↩

2. In the visible text, the reference number appears as 12. ↩