Two spherical bodies of same materials having radii 0.2 m and 0.8 m are placed in same atmosphere. The temperature of the smaller body is 800 K and temperature of the bigger body is 400 K . If the energy radiated from the smaller body is E, the energy radiated from the bigger body is (assume, effect of the surrounding temperature to be negligible),

<p>To determine the energy radiated by two spherical bodies, both made of the same material but with different radii and temperatures, we can use the Stefan-Boltzmann law. This law states that the energy radiated by a black body per unit time is proportional to the product of the surface area $ A $ and the fourth power of its temperature $ T $.</p> <p>The formula for power $ P $ is given by:</p> <p>$ P = \sigma e A T^4 $</p> <p>Where:</p> <p><p>$ \sigma $ is the Stefan-Boltzmann constant</p></p> <p><p>$ e $ is the emissivity of the material</p></p> <p><p>$ A $ is the surface area of the sphere</p></p> <p><p>$ T $ is the temperature in Kelvin</p></p> <p>Expressing this relationship as a proportionality:</p> <p>$ P \propto A T^4 $</p> <p>For a sphere, the surface area $ A $ is:</p> <p>$ A = 4\pi R^2 $</p> <p>Hence, the power radiated is proportional to:</p> <p>$ P \propto R^2 T^4 $</p> <p>For two bodies with radii $ R_1 $ and $ R_2 $, and temperatures $ T_1 $ and $ T_2 $, the ratio of their powers is:</p> <p>$ \frac{P_1}{P_2} = \left( \frac{R_1}{R_2} \right)^2 \left( \frac{T_1}{T_2} \right)^4 $</p> <p>Given:</p> <p><p>$ R_1 = 0.2 \, \text{m} $, $ T_1 = 800 \, \text{K} $</p></p> <p><p>$ R_2 = 0.8 \, \text{m} $, $ T_2 = 400 \, \text{K} $</p></p> <p>Substituting into the formula:</p> <p>$ \frac{E}{E'} = \left( \frac{0.2}{0.8} \right)^2 \left( \frac{800}{400} \right)^4 $</p> <p>Calculating each part:</p> <p><p>$\left( \frac{0.2}{0.8} \right)^2 = \left( \frac{1}{4} \right)^2 = \frac{1}{16}$</p></p> <p><p>$\left( \frac{800}{400} \right)^4 = 2^4 = 16$</p></p> <p>Combining these calculations:</p> <p>$ \frac{E}{E'} = \frac{1}{16} \times 16 = 1 $</p> <p>This implies that the energy radiated by the bigger body $ E' $ equals the energy radiated by the smaller body $ E $. Thus, $ E' = E $.</p>