Abstract
It is shown that, when account is taken of the temperature dependence of the electrical conductivity of a medium in steady-state conditions, the electrical potential obeys a quasi-linear, second-order partial differential equation. The equation is shown to be easily solved by means of a generalized Kirchhoff transformation, giving coordinate-free solutions in terms of functions which obey Laplace's equation. Steady-state temperatures resulting from the potential are shown to be significantly influenced by the quasi-linear potential when compared with the expected form which assumes the potential to satisfy Laplace's equation.
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Young, J.H. Quasi-linear electrical potentials in steady-state Joule heating. J Eng Math 21, 33–40 (1987). https://doi.org/10.1007/BF00127691
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DOI: https://doi.org/10.1007/BF00127691