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A thermally insulated piece of metal is heated under atmosphere by an electric current so that it receives electric energy at a constant power P. This leads to an increase of the absolute temperature T of the metal with time t as follow $T = a t^{1 / 4}$ then, the heat capacity C_P is

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$\frac{4 P T^{3}}{a^{4}}$

$\frac{4 P T^{2}}{a^{3}}$

None of these

answer is A.

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Detailed Solution

$d H = P d t$

$C_{p} = \frac{P d t}{d T}$

$H e r e,$ $T = a t^{1 / 4}$ Or $\frac{T}{a} = t^{1 / 4} \Rightarrow t = \frac{T^{4}}{a^{4}}$

$\frac{d t}{d T} = \frac{1}{a^{4}} (4 T^{3})$

$C_{P} = \frac{P d t}{d T} = \frac{4 P T^{3}}{a^{4}}$

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