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    "title": "Johnson–Nyquist electrical thermal noise: measurement and equilibrium theory",
    "summary": "An electrical conductor can produce measurable random voltage even when no signal is being sent through it. Johnson measured this thermal noise and investigated how it depends on resistance, temperature and the measuring circuit. Nyquist gave an equilibrium explanation: in the classical regime, a resistor has voltage mean-square contribution 4kTR per unit ordinary-frequency bandwidth. The two separate papers connect an observed limit of sensitive electronics with thermal statistical physics.",
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      "statement": "# Johnson–Nyquist electrical thermal noise: measurement and equilibrium theory\n\n## A quiet resistor is not perfectly quiet\n\nA 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.\n\n## Bandwidth is part of the answer\n\nIn 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.\n\n## Estimate a classroom noise floor\n\nUse 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.\n\n## Keep the originals and later theory apart\n\nThat 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.\n\n## Build a checkable explanation\n\nHave 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.\n\n## Scientific scope\n\nThis 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.\n\n## Sources for this explanation\n\n- [Thermal Agitation of Electricity in Conductors](https://facultystaff.richmond.edu/~olipan/Johnson.pdf): 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.\n- [Thermal Agitation of Electric Charge in Conductors](https://123.physics.ucdavis.edu/johnson_files/nyquist_1928.pdf): 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.\n",
      "summary": "An electrical conductor can produce measurable random voltage even when no signal is being sent through it. Johnson measured this thermal noise and investigated how it depends on resistance, temperature and the measuring circuit. Nyquist gave an equilibrium explanation: in the classical regime, a resistor has voltage mean-square contribution 4kTR per unit ordinary-frequency bandwidth. The two separate papers connect an observed limit of sensitive electronics with thermal statistical physics.",
      "title": "Johnson–Nyquist electrical thermal noise: measurement and equilibrium theory"
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    "what_changed": "An electrical conductor can produce measurable random voltage even when no signal is being sent through it. Johnson measured this thermal noise and investigated how it depends on resistance, temperature and the measuring circuit. Nyquist gave an equilibrium explanation: in the classical regime, a resistor has voltage mean-square contribution 4kTR per unit ordinary-frequency bandwidth. The two separate papers connect an observed limit of sensitive electronics with thermal statistical physics.",
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        "title": "A quiet resistor is not perfectly quiet",
        "text": "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."
      },
      {
        "title": "Bandwidth is part of the answer",
        "text": "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."
      },
      {
        "title": "Estimate a classroom noise floor",
        "text": "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."
      },
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