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I derived an expression for the time constant of the temperature of an electrical conductor and was looking for some constructive feedback.

Initially there is no current flowing through the conductor and its temperature (assume its isotropic) is equal to ambient. Then we start flowing some current through the conductor and in a first order fashion it settles at some final temperature.

Because the response looks like a first order response, I want to make a first order approximation. I can estimate the initial rate of temp rise by assuming an adiabatic conductor. I will divide the heat generation (I²R) by its heat capacity and get the max/initial temp rate of change, which is proportional to the square of the current density.

The time constant of the conductor is going to be the temp rise divided by initial temp rate of change.

What do you think?

enter image description here

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    \$\begingroup\$ Your assumptions appear to be similar to the assumptions for Onderdonk's equation for fusing current, electronics.stackexchange.com/questions/580286/… . Onderdonk's equation is used to determine how long a copper wire can survive (i.e. not melt) for a given current, assuming the wire is perfectly insulated. The assumption of perfectly insulated works well cases where the time scale is short, like a current pulse induced by lightning. \$\endgroup\$
    – C. Dunn
    Commented Sep 12, 2023 at 16:54

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Your analysis looks accurate, and makes physical sense.

I would just tweak the final bullet from:

"Temperature rise proportional to square of current density"

to

"Rate of temperature rise is proportional to square of current density"

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    \$\begingroup\$ Thanks for that. \$\endgroup\$
    – user50655
    Commented Sep 12, 2023 at 17:22

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