Two metals P and Q of length l1 and l2 are heated through the same temperature difference. If the ratio of the…

Analyze the question

Both metals are heated through the same temperature difference (Δθ). We are given the ratio of their linear expansivities (α) and the ratio of their original lengths, and we need the ratio of their increases in length.

Concept

Linear expansion is given by:

ΔL = α · L · Δθ

So for the same Δθ, the increase in length is proportional to α × L.

Solution

• α_P : α_Q = 2 : 3
• L_P : L_Q = 3 : 4

ΔL_P / ΔL_Q = (α_P × L_P × Δθ) / (α_Q × L_Q × Δθ)

= (2 × 3) / (3 × 4)

= 6 / 12

= 1/2

So the ratio of the increase in length of P to Q is 1 : 2.

Answer

Tip

When a question says "same temperature difference," Δθ cancels out. Then just multiply the ratios straight across (α ratio × length ratio). Watch out for the common mistake of adding the ratios (2+3, 3+4), which gives the wrong options like 5:7 or 7:5.