= Solution
Thermal agitation in a <resistor> produces \b[<Johnson-Nyquist noise>], even without an applied mean current. In the classical low-frequency regime, its one-sided open-circuit voltage spectrum is $S_V=4k_BTR$ and its equivalent short-circuit current spectrum is $S_I=4k_BT/R$. In an <Effective noise bandwidth> $B$,
$$
\boxed{v_{\rm rms}=\sqrt{4k_BTRB},\qquad i_{\rm rms}=\sqrt{4k_BTB/R}.}
$$
For a matched load, available thermal noise power is $k_BTB$. A <resistor> and load at the same <temperature> exchange noise power without a net heat flow.
Cooling reduces the rms amplitude as $\sqrt T$ if <Electrical resistance> and <bandwidth> remain fixed. A cryogenic bath or refrigerator, <thermal anchoring> and controlled heat input can maintain a steady low <temperature>. Radiation shields, an insulating <vacuum>, thermally anchored leads and limited measurement dissipation reduce heat leaks and self-heating. <Temperature> sensing and stable <feedback control> reduce slow temperature-induced drift; keeping <temperature> constant does not remove microscopic <thermal fluctuations>.
\Image[/past-exam-of-the-physics-a-course-of-the-university-of-cambridge/2016/paper-1-thermal-noise.png]
{title=At fixed resistance and bandwidth, thermal-noise power scales with temperature and rms amplitude with its square root}
{source=paper-1-thermal-noise.py}
<Bandwidth> restriction, suitable impedance and a low-noise amplifier complement cooling. Warm leads, contacts and the amplifier can otherwise dominate the observed noise. At fixed mean current, <shot noise> is not generally reduced merely by cooling, and low-frequency excess noise need not follow the Johnson law. At frequencies with $hf\gtrsim k_BT$, quantum noise expressions replace the classical approximation.<Op-amp Noise by Analog Devices>[Distinction between resistor thermal noise, amplifier noise and current-noise contributions]{ref}
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