A brief pedagogic review of frequency-dependent Johnson Nyquist noise

We present a pedagogical and self-contained derivation of Johnson–Nyquist formula that relates the voltage fluctuations due to the thermal noise to the impedance in linear electrical circuits using linear response theory. Spectral densities and measurement bandwidth are carefully defined following Nyquist’s original formulation. Explicit expressions are derived for voltage fluctuations across a resistor in R, RC (series and parallel), and LCR circuits. Reactive elements are shown to shape but not generate noise, producing frequency-dependent Johnson–Nyquist relations governed by circuit susceptibility. We demonstrate that while pure resistors exhibit frequency-independent (white) Johnson–Nyquist noise, circuits containing reactive elements can display frequency-dependent colored noise spectra. Specifically, we analyze pure resistors, series RC circuits, parallel RC circuits, and series LCR resonators, showing how circuit topology determines whether the noise spectrum is white or colored. We explain the physical origin of frequency dependence through the concept of effective resistance and current redistribution at different frequencies. Analogies with the generalized Langevin equation are explained.