A sphere of surface area $4 {\mathrm{m}}^2$ at temperature $400 \mathrm{K}$ and having emissivity $0.5$ is located in an environment of temperature $200 \mathrm{K}$. The net rate of energy exchange of the sphere is

(Stefan Boltzmann constant $\sigma =5.67\times {10}^{-8} \mathrm{W} {\mathrm{m}}^{-2} {\mathrm{K}}^4$)

The filament of a light bulb has surface area $64 {\mathrm{mm}}^2.$ The filament can be considered as a black body at temperature $2500 \mathrm{K}$ emitting radiation like a point source when viewed from far. At night the light bulb is observed from a distance of $100 \mathrm{m}$. Assume the pupil of the eyes of the observer to be circular with radius $3 \mathrm{mm}$. Then:

(Take Stefan-Boltzmann constant $=5.67\times {10}^{-8} \mathrm{W} {\mathrm{m}}^{-2} {\mathrm{K}}^{-4},$ Wiens' displacement constant $=2.90\times {10}^{-3} \mathrm{m} \mathrm{K}$, Planck's constant $=6.60\times {10}^{-34} \mathrm{J} \mathrm{s}$, speed of light in vacuum $=3.00\times {10}^8 \mathrm{m}{\mathrm{s}}^{-1}$)

Two black bodies $A$ and $B$ have equal surface areas and are maintained at temperatures $27°\mathrm{C}$ and $177°\mathrm{C}$ respectively. What will be the ratio of the thermal energy radiated per second by $A$ to that by $B?$

A black coloured solid sphere of radius $R$ and mass $M$ is inside a cavity with a vacuum inside. The walls of the cavity are maintained at temperature $T_0$. The initial temperature of the sphere is $3T_0$. If the specific heat of the material of the sphere varies as $\alpha T^3$ per unit mass with the temperature $T$ of the sphere, where $\alpha$ is a constant, then the time taken for the sphere to cool down to temperature $2T_0$ will be

($\sigma$ is Stefan Boltzmann constant)