Question 68: Thermodynamics & KTG

Question

Two black bodies $$\mathrm{P}$$ and $$\mathrm{Q}$$ have equal surface areas and are kept at temperatures $$127^{\circ} \mathrm{C}$$ and $$27^{\circ} \mathrm{C}$$ respectively. The ratio of thermal power radiated by A to that by B is

Accepted Answer

The thermal power radiated by a black body is given by Stefan's law, according to which the power radiated (P) is directly proportional to the fourth power of the absolute temperature (T) of the body. This can be mathematically expressed as:
$$P = \sigma e A T^4$$
where:
$$\sigma$$ is the Stefan-Boltzmann constant,
$$e$$ is the emissivity of the surface,
$$A$$ is the surface area,
$$T$$ is the absolute temperature in Kelvin.
For the given problem, since both black bodies have equal surface areas and are perfect black bodies (assuming $$e = 1$$ for both), the ratio of their thermal powers $$P_\mathrm{P}$$ to $$P_\mathrm{Q}$$ can be simplified to the ratio of the fourth powers of their absolute temperatures. To work with absolute temperatures, we must first convert the given Celsius temperatures to Kelvin:
Temperature of $$\mathrm{P} = 127^{\circ} \mathrm{C} = 127 + 273 = 400\ \mathrm{K}$$
Temperature of $$\mathrm{Q} = 27^{\circ} \mathrm{C} = 27 + 273 = 300\ \mathrm{K}$$
The ratio of their thermal powers can thus be expressed as:
$$\frac{P_\mathrm{P}}{P_\mathrm{Q}} = \frac{\sigma 	imes A 	imes (400)^4}{\sigma 	imes A 	imes (300)^4}$$
Simplifying this, we get:
$$\frac{P_\mathrm{P}}{P_\mathrm{Q}} = \frac{(400)^4}{(300)^4} = \left(\frac{4}{3}ight)^4$$
Calculating the power of 4:
$$\frac{P_\mathrm{P}}{P_\mathrm{Q}} = \left(\frac{4^4}{3^4}ight) = \frac{256}{81}$$
Therefore, the correct option is:
Option D $$256: 81$$.

Answer

$$81: 256$$

Answer

$$177: 127$$

Answer

$$127: 177$$

Answer

$$256: 81$$

Thermodynamics & KTG

Two black bodies $\mathrm{P} and \mathrm{Q} have equal surface areas and are kept at temperatures 127^{\circ} \mathrm{C} and 27^{\circ} \mathrm{C}$ respectively. The ratio of thermal power radiated by A to that by B is

Choose the correct answer: