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System:

A small metal bar with length l=2.5mm, width w=0.2mm, thickness t=0.2μm with resistance R is on a substrate, with the rest of the material open to the air.

I want to work out what temperature the bar will be with a given current I flowing through it (and conversely what current is needed to maintain it at a temperature T).

My thoughts are to balance the power in and out of the bar.

I've got

$$P_{in}=I^2R$$

$$P_{out}=P_{blackbody} + P_{air} + P_{substrate}$$

I can calculate the blackbody radiation power fine (using an emissivity of 0.1), but I'm not sure how to calculate the conduction into the substrate and air.

I only need an estimate, so I was thinking of ignoring the effects of convection, and just treating the air and the substrate as heat reservoirs.

This is looking at short timescales (less than 1 second really), so I was thinking of ignoring heating of the air and substrate, assuming that they stay at room temperature.

My main problem comes from the fact that Fourier's law, q=-kdT/dx requires a temperature gradient, which I have no idea how to calculate. Approximating the gradient as ΔT/Δx requires some characteristic length scale, which I'm not sure how to approach.

Does anyone have any suggestions on how I can estimate this, or point out some flaw in my reasoning (bearing in mind that this is only a rough estimate)?