The ratio of lengths to radii of two rods of same material is 1 : 2 and 2 :

3. If temperature difference between two ends of the rod is same, then what will be the ratio of heat flow ?

Question

The ratio of lengths to radii of two rods of same material is 1 : 2 and 2 : 3. If temperature difference between two ends of the rod is same, then what will be the ratio of heat flow ?

UnlockIAS · Accepted Answer

Correct — option (4), 8 : 9. The rate of heat flow (heat current) through a rod under steady-state conduction is H = kA(delta T)/L, where k is thermal conductivity, A is the cross-sectional area, delta T is the temperature difference across the ends, and L is the length. Both rods are of the 'same material', so k is identical for both and cancels out of the ratio; the temperature difference is stated to be the same for both rods, so delta T also cancels. What remains is H is proportional to A/L, and since each rod is circular in cross-section, A = pi r-squared, so H is proportional to (r-squared)/L. The question gives the ratio of length to radius for each rod (rod 1, L:r = 1:2; rod 2, L:r = 2:3), which is equivalent to saying rod 1's length and radius are in proportion 1 and 2 respectively (in some common unit), and rod 2's are in proportion 2 and 3. Substituting, H1/H2 = (r1-squared / L1) divided by (r2-squared / L2) = (r1-squared times L2) / (L1 times r2-squared) = (2-squared times 2) / (1 times 3-squared) = (4 times 2) / (1 times 9) = 8/9. So the ratio of heat flow, H1 : H2, is 8 : 9, matching option (4) exactly.

Why the other options are wrong — (a) This value does not follow from H proportional to r-squared/L for the given ratios; it looks like it could result from comparing only the radii (2:3, inverted to 3:2) while ignoring the length term and the squaring of the radius entirely. Heat flow depends on the cross-sectional area, not the radius directly, so radius must be squared before it enters the ratio, and the length term cannot be dropped.  (b) This value also fails to track H proportional to r-squared/L correctly; it is consistent with squaring only one rod's radius, or with using the length ratio directly as the answer without incorporating the area term at all. Both the length and the squared radius of each rod must be combined multiplicatively as A/L, not compared piecemeal.  (c) This value would follow only from comparing lengths alone (1:2, simplified or misapplied) or from some other partial combination that ignores the area (r-squared) contribution altogether. Since heat flow is proportional to cross-sectional area divided by length, leaving out the r-squared term or mishandling the ratio direction produces a value that does not match the correct 8:9 result.

Concept: Steady-state heat conduction through a rod follows Fourier's law in its simplest one-dimensional form: the rate of heat flow H = kA(delta T)/L, where k is the material's thermal conductivity, A the cross-sectional area, delta T the temperature difference maintained across the two ends, and L the length of the rod. For rods of the same material carrying the same temperature difference, H is proportional to A/L — a shorter, fatter rod conducts heat faster than a longer, thinner one, which is the same geometric logic used for electrical resistance (R = rho L/A) except heat flow is proportional to A/L directly rather than to its reciprocal, since H plays the role of a current driven by a temperature 'potential difference' across a thermal 'resistance' L/(kA).

State PCS 2025 Q51 — Science & Technology PYQ with Answer

State PCS 2025 Q51 — Science & Technology PYQ with Answer