When geometry is irrelevant for heat diffusion: the transition from lumped-element to field formulations

Abstract

Thermal systems are an attractive setting for exploring the connections between the lumped-element approximations of elementary circuit theory and the partial-differential field equations of mathematical physics. In a calculation suitable for an undergraduate course in mathematical physics, we show that the response function between an oscillating heater and temperature probe has a smooth crossover between a low-frequency, ‘lumped-element’ regime where the system behaves as a whole and a high-frequency regime dominated by the spatial dependence of the temperature field. Undergraduates can also easily (and cheaply) explore these ideas experimentally in a typical advanced laboratory course. Because the characteristic frequencies are low, ≈0.03 Hz, measuring the response frequency by frequency is slow and challenging; to speed up the measurements, we introduce a useful, if underappreciated, experimental technique based on a multisine power signal that sums carefully chosen harmonic components with random phases. Strikingly, a simple model assuming a one-dimensional, finite rod predicts a temperature response in the frequency domain that closely approximates experimental measurements from an irregular, blob-shaped object. A spherical model gives similar results. Thus, the frequency response of this irregular thermal system has surprisingly little sensitivity to its geometry, an example of—and justification for—the toy models so beloved of physicists.