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One end of a thermally insulated rod is kept at a temperature T₁ and the other at T₂. The rod is composed of two sections joined end to end, of lengths l₁ and l₂ and thermal conductivities K₁ and K₂ respectively, the section of conductivity K₁ being the one at T₁. The temperature at the interface of the two sections is

Asked in AIEEE 2007 · Junction temperature

• (1) (K₁l₁T₁+K₂l₂T₂)/(K₁l₁+K₂l₂)
• (2) (K₂l₂T₁+K₁l₁T₂)/(K₁l₁+K₂l₂)
• (3) (K₂l₁T₁+K₁l₂T₂)/(K₂l₁+K₁l₂)
• (4) (K₁l₂T₁+K₂l₁T₂)/(K₁l₂+K₂l₁)

Step-by-step solution

Why the other options are wrong

• (1) comes from weights each end temperature by the product Kl of its own section. Conductance goes as K/l, so a longer section should weigh less, not more; this option makes a long section pull the interface toward itself.
• (2) comes from uses Kl again but with the two products swapped over. It happens to give the right sense for the length but the wrong one for the conductivity, so it fails the simple test l₁=l₂, K₁=9K₂, where the interface must sit near the K₁ end.
• (3) comes from pairs each conductivity with the other section's length, which is what comes of writing (K₁(T₁-T))/(l₁)=(K₂(T-T₂))/(l₂) and then cross-multiplying the wrong diagonal.

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