Two rods of different materials have lengths ' ₁ ' and ' ₂ ' whose coefficient of linear expansions are ' ₁ ' and ' ₂ ' respectively. If the difference between the two lengths is independent of temperature then

Step-by-step solution

Let the initial rod lengths at reference temperature T₀ be ₁ and ₂ , with linear expansion coefficients ₁ and ₂ . Their expanded lengths at temperature T are ₁' = ₁(1 + ₁ T) and ₂' = ₂(1 + ₂ T) . The difference ₁' - ₂' = ( ₁ - ₂) + ( ₁ ₁ - ₂ ₂) T . For this difference to be temperature-independent, the coefficient of T must vanish, giving ₁ ₁ = ₂ ₂ . This implies ₁ ₂ = ₂ ₁ , which corresponds to option B. Final answer: B