A quiet resistor is not perfectly quiet
A resistor connected to a sufficiently sensitive circuit produces fluctuating voltage even without a deliberately applied signal. Johnson investigated this electrical agitation experimentally; Nyquist supplied an equilibrium explanation in a separate paper. Their linked contribution connects random microscopic motion with an observable electrical signal. Random here does not mean anything can happen without a pattern: the average squared voltage follows a quantitative rule under stated temperature, resistance and bandwidth conditions.
Bandwidth is part of the answer
In the familiar low-frequency equilibrium limit, the one-sided voltage noise density per hertz is four times Boltzmann constant times absolute temperature times resistance. Integrating a flat density over a measurement bandwidth B gives mean-square voltage 4 k T R B. Mean-square is an average of squared readings; take its square root to obtain the root-mean-square voltage. A one-sided ordinary-frequency convention is essential. Changing to angular frequency or a different spectral convention changes the notation and factors.
Estimate a classroom noise floor
Use a fictional flat-band example: resistance 1000 ohms, temperature 300 kelvin and bandwidth 1000 hertz. With the modern exact SI Boltzmann constant 1.380649 times ten to the minus 23 joules per kelvin, the mean-square voltage is 1.6567788 times ten to the minus 14 volts squared. Its square root is about 0.129 microvolts. Doubling bandwidth doubles the mean square, while the root-mean-square voltage increases only by the square root of two, about 1.414.
Keep the originals and later theory apart
That arithmetic uses a modern constant, not Johnson historical estimate. The original experiment had substantial uncertainty. Nyquist included a quantum thermal factor, but it is not interchangeable with later zero-point or symmetrized formulas. Equilibrium, dissipation and the actual circuit response remain necessary. Non-ohmic devices, externally driven systems, shot noise and other noise mechanisms cannot all be assigned the simple resistor formula merely because their output fluctuates.
Build a checkable explanation
Have your agent produce a unit-checked worksheet with resistance, temperature, bandwidth, spectral convention and output units visible. Ask it to repeat the calculation after narrowing the filter, then explain the square-root change without claiming the resistor became colder. Preserve Johnson experimental byline and Nyquist theoretical byline separately while linking the shared family. A useful submission can flag an incorrect unit or convention; it does not need to manufacture a new laboratory measurement.
Scientific scope
This is one observation-and-explanation family with two original bylines, not a joint paper. Equilibrium, dissipative impedance, bandwidth, filtering and one-sided versus angular-frequency conventions matter. The originals exclude unqualified material independence for non-ohmic, thermionic and photoelectric devices. Johnson measured a historical Boltzmann estimate with substantial uncertainty, not the modern exact SI value. Nyquist original quantum correction uses hf/(exp(hf/kT)-1), not a later zero-point/coth convention. Schottky earlier suggestion and acknowledged criticism and measurement assistance remain distinct. No general statement about all noise or every zero-temperature observable follows.
Read the original work
Sources and reading locations for this explanation
- Thermal Agitation of Electricity in Conductors
Johnson, Physical Review32,97–109, selected original pages; Nyquist,32,110–113, all four original pages. Separate bylines, historical measurement uncertainty and equilibrium/quantum conventions retained.
- Thermal Agitation of Electric Charge in Conductors
Johnson, Physical Review32,97–109, selected original pages; Nyquist,32,110–113, all four original pages. Separate bylines, historical measurement uncertainty and equilibrium/quantum conventions retained.
This English explanation is an editorial reading view. It does not replace the complete original work or certify its science.
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People and systems
J. B. Johnson; H. Nyquist
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