Past exam of the physics A course of the University of Cambridge / 2016 / Paper 1 / Section D / D13 / b / Solution

Thermal agitation in a resistor produces Johnson-Nyquist noise, even without an applied mean current. In the classical low-frequency regime, its one-sided open-circuit voltage spectrum is $S_V=4k_{B}TR$ and its equivalent short-circuit current spectrum is $S_I=4k_{B}T/R$. In an Effective noise bandwidth $B$,For a matched load, available thermal noise power is $k_{B}TB$. 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.

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 $hf\gtrsim k_{B}T$, quantum noise expressions replace the classical approximation.^[ref]