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Two rods of lengths l1 and l2 with thermal conductivities x1 and x2 are joined end to end (lateral surfaces insulated). Find the thermal conductivity of a single uniform rod of length l1 + l2 that would conduct heat at the same rate.

1. (x1*l1 + x2*l2)/(l1 + l2)
2. (l1 + l2)/(l1/x1 + l2/x2)
3. (x1*x2)/(x1 + x2)
4. (x1*l2 + x2*l1)/(l1 + l2)

Correct answer: (l1 + l2)/(l1/x1 + l2/x2)

Solution

Thermal resistances add in series, so (l1 + l2)/x = l1/x1 + l2/x2, giving x = (l1 + l2)/(l1/x1 + l2/x2).

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