From rods to blobs: When geometry is irrelevant for heat diffusion

TL;DR

This paper demonstrates that the frequency response of irregular thermal objects can be effectively modeled by simple geometric approximations, bridging circuit theory and heat diffusion physics in an undergraduate setting.

Contribution

It introduces an experimental technique using multisine signals to efficiently measure thermal response and shows that geometry has minimal impact on frequency response in certain thermal systems.

Findings

The response function exhibits a crossover from lumped-element to spatially dependent behavior.

A simple rod model closely predicts the response of irregular objects.

Frequency response is nearly independent of the system's geometry.

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, a topic that has been neglected in physics curricula. 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 an electrical capacitor and a high-frequency regime dominated by the spatial dependence of the temperature field. Undergraduates can easily (and cheaply) explore these ideas experimentally in a typical advanced laboratory course. Because the characteristic frequencies are low, ($\approx$ 30 s)$^{-1}$, 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, we find that the simple model of a one-dimensional, finite rod predicts a temperature response in the frequency domain that closely approximates experimental measurements from an irregular, blob-shaped object. The unexpected conclusion is that the frequency response of this irregular thermal system is nearly independent of its geometry, an example of---and justification for---the "spherical cow" approximations so beloved of physicists.