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Two different wires having lengths L₁ and L₂, and respective temperature coefficients of linear expansion α₁ and α₂, are joined end-to-end. Then the effective temperature coefficient of linear expansion is

Asked in JEE Main 5th Sept 2nd Shift 2020 · Linear expansion and composite rods

• (1) (α₁L₁+α₂L₂)/(L₁+L₂)
• (2) 2√α₁α₂
• (3) (α₁+α₂)/2
• (4) 4(α₁α₂)/(α₁+α₂)(L₂L₁)/((L₂+L₁)²)

Step-by-step solution

Why the other options are wrong

• (2) comes from a geometric-mean guess; it is not even dimensionally sensible as a weighted result, since the two lengths — which decide how much each wire contributes — never appear in it.
• (3) comes from is the plain average, which is the special case L₁=L₂ of the correct expression. Using it in general gives a long, low-α wire the same say as a short, high-α one.
• (4) comes from borrows the harmonic-mean form used for resistors in parallel or lenses in contact; expansions of parts joined end to end simply add, so nothing reciprocal belongs here.

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