### Thermal Agitation of Electricity in Conductors

#### Abstract

Statistical fluctuation of electric charge exists in all conductors, producing random variation of potential between the ends of the conductor. The effect of these fluctuations has been measured by a vacuum tube amplifier and thermocouple, and can be expressed by the formula ${\overline{I}}^{2}=\left(\frac{2kT}{\pi }\right)\int {0}^{\infty }R\left(\omega \right){|Y\left(\omega \right)|}^{2}d\omega$. $I$ is the observed current in the thermocouple, $k$ is Boltzmann's gas constant, $T$ is the absolute temperature of the conductor, $R\left(\omega \right)$ is the real component of impedance of the conductor, $Y\left(\omega \right)$ is the transfer impedance of the amplifier, and $\frac{\omega }{2\pi }=f$ represents frequency. The value of Boltzmann's constant obtained from the measurements lie near the accepted value of this constant. The technical aspects of the disturbance are discussed. In an amplifier having a range of 5000 cycles and the input resistance $R$ the power equivalent of the effect is $\frac{{\overline{V}}^{2}}{R}=0.8×{10}^{-16}$ watt, with corresponding power for other ranges of frequency. The least contribution of tube noise is equivalent to that of a resistance ${R}_{c}=1.5×\frac{{10}^{5}{i}_{p}}{\mu }$, where ${i}_{p}$ is the space current in milliamperes and $\mu$ is the effective amplification of the tube.

DOI: http://dx.doi.org/10.1103/PhysRev.32.97

• Published in the issue dated July 1928

© 1928 The American Physical Society

#### Authors & Affiliations

J. B. Johnson

• Bell Telephone Laboratories, Incorporated

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