JEE Advanced 2021 Paper 1 · Q19 · Heat Radiation / Stefan-Boltzmann

Question

A small object is placed at the center of a large evacuated hollow spherical container. Assume that the container is maintained at 0 K. At time t = 0, the temperature of the object is 200 K. The temperature of the object becomes 100 K at t = t₁ and 50 K at t = t₂. Assume the object and the container to be ideal black bodies. The heat capacity of the object does not depend on temperature. The ratio (t₂/t₁) is ___.

SolveKar AI · Accepted Answer

Heat balance with surroundings at 0 K: −C(dT/dt) = σA·T⁴ ⇒ dT/T⁴ = −(σA/C)dt. Integrate from 200 to 100 K: [1/(100)³ − 1/(200)³]/3 = (σA/C)t₁. From 200 to 50 K: [1/(50)³ − 1/(200)³]/3 = (σA/C)t₂. Ratio t₂/t₁ = [1/50³ − 1/200³]/[1/100³ − 1/200³] = [8/10⁶ − 1/(8·10⁶)]/[1/10⁶ − 1/(8·10⁶)] = (64 − 1)/(8 − 1) = 63/7 = 9.

JEE Advanced 2021 Paper 1 · Q19 · Heat Radiation / Stefan-Boltzmann

Solution