Two metals P and Q of length l1 and l2 are heated through the same temperature difference. If the ratio of the linear expansivities of P to Q is 2:3 and the...

Two metals P and Q of length l1 and l2 are heated through the same temperature difference. If the ratio of the linear expansivities of P to Q is 2:3 and the ratio of lengths is 3:4. What is the ratio of increase in length of P to Q?

Answer Details

When two different materials are heated through the same temperature difference, the length of the materials changes. This change in length due to temperature is called thermal expansion. The amount of thermal expansion depends on the material's coefficient of linear expansion, which is denoted by alpha (α). From the problem, we know that the ratio of the linear expansivities of P to Q is 2:3, which means that alpha of P is (2/3) times alpha of Q. Also, the ratio of the lengths of P to Q is 3:4. Let's assume the original lengths of P and Q are 3x and 4x respectively. When heated, the increase in length of P will be α_P × l₁ = (2/3)α_Q × l₁ = (2/3)α_Q × (3x), since length of P is 3x and the ratio of linear expansivities is 2:3. Similarly, the increase in length of Q will be α_Q × l₂ = α_Q × (4x), since length of Q is 4x. Now, to find the ratio of increase in length of P to Q, we divide the increase in length of P by the increase in length of Q: [(2/3)α_Q × (3x)] / [α_Q × (4x)] = (2/3) × (3/4) = 1/2 Therefore, the ratio of increase in length of P to Q is 1:2. Hence, the answer is (C) 1:2.