Question

Spheres $P$ and $Q$ are uniformly constructed from the same material which is a good conductor of heat and the radius of $Q$ is thrice the radius of $P$. The rate of fall of temperature of $P$ is $x$ times that of $Q$. When both are at the same surface temperature, the value of $x$ is :

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Correct option is C)

$dtdT =−ms4σeAT_{0} (T−T_{0})=−(ρ4/3πr_{3})s4σe(4πr_{2})T_{0} (T−T_{0})$

Thus, $dtdT ∝r1 $

$(dtdT )_{Q}(dtdT )_{P} =r_{P}r_{Q} =r_{P}3r_{P} =3$

$∴(dtdT )_{P}=3×(dtdT )_{Q}$

Thus, $dtdT ∝r1 $

$(dtdT )_{Q}(dtdT )_{P} =r_{P}r_{Q} =r_{P}3r_{P} =3$

$∴(dtdT )_{P}=3×(dtdT )_{Q}$

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