The total energy of a black body radiation source is collected for five minutes and used to heat water. The temperature of the water increases from $10.0^{\circ} C$ to $11.0^{\circ} C$. The absolute temperature of the black body is doubled and its surface area halved and the experiment repeated for the same time. Which of the following statements would be most nearly correct?
KVPY 2012, Diffcult
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(b)
Energy radiated from a black body is given by
$U=\sigma \cdot A \cdot T^4 \cdot t$
where, $\sigma=$ Stefan's constant, $A=$ area, $T=$ absolute temperature and $t=$ time.
Now, ratio of energy collected in two given cases is
$\frac{U_2}{U_1} =\frac {(A / 2)(2 T)^4} {A T^4}=8$
Hence, ratio of temperature rise of water is
$\frac{m s \Delta T_2}{m s \Delta T_1}=\frac{U_2}{U_1}=8 .$
$\Rightarrow \quad \Delta T_2=8 \Delta T_1$
$\text { As, } \Delta T_1=1^{\circ} C \text { and } \Delta T_2=8^{\circ} C .$
So, temperature of water increases from $10^{\circ} C$ to $18^{\circ} C .$
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