MPSC 2025 Prelims Paper-I · Question 51 of 100

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.

• (1)3 : 2 — 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.
• (2)4 : 3 — 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.
• (3)1 : 3 — 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.

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).

MPSC's physics questions frequently give two rods (or wires) differing only in length and radius and ask for a ratio — of heat flow, resistance, or resistance-per-unit-length — testing whether a candidate can correctly substitute a geometric ratio into the governing proportionality and handle the squaring of the radius when area is involved. The recurring trap is forgetting that area depends on radius-squared, not radius, which silently changes the answer if skipped.

• Steady-state heat flow through a rod: H = kA(delta T)/L, with k the thermal conductivity, A the cross-sectional area, delta T the temperature difference, and L the length.
• For rods of the same material and same temperature difference, H is proportional to A/L, i.e. proportional to (radius-squared)/length for a circular cross-section.
• This is the thermal analogue of electrical conduction, where current is proportional to a driving potential difference divided by a resistance that itself depends on the conductor's geometry.

• Using the radius ratio directly in the heat-flow formula instead of squaring it to get the cross-sectional area
• Confusing the heat-conduction proportionality (H proportional to A/L) with the electrical-resistance proportionality (R proportional to L/A), which are geometric inverses of each other
• Mixing up which ratio belongs to which rod when the question states two separate length-to-radius ratios for two different rods

MPSC's physics section regularly poses ratio problems on heat conduction, electrical resistance, or terminal velocity that hinge on correctly substituting given length and radius ratios into a proportionality and remembering where a quantity must be squared or cubed.

No directly related past PYQ was found.

• practice — not a real PYQ

  Two rods of the same material have radii in the ratio 1 : 2 and lengths in the ratio 1 : 1. If the temperature difference across each rod is the same, what is the ratio of the rate of heat flow through them ?

  • (a)1 : 2
  • (b)1 : 4
  • (c)2 : 1
  • (d)4 : 1

  Answer(b) 1 : 4 — since H is proportional to (radius-squared)/length and the lengths are equal, H1:H2 = (1-squared):(2-squared) = 1:4.

• practice — not a real PYQ

  The formula for the steady-state rate of heat conduction through a uniform rod of cross-sectional area A, length L, thermal conductivity k, and end-to-end temperature difference delta T is ?

  • (a)H = kA(delta T)/L
  • (b)H = kL(delta T)/A
  • (c)H = k(delta T)/(AL)
  • (d)H = kAL(delta T)

  Answer(a) H = kA(delta T)/L — the standard one-dimensional steady-state conduction equation, directly proportional to area and temperature difference and inversely proportional to length.