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Dimensional Analysis

A solid at te...

Question

A solid at temperature $T_{1}$ is kept in an evacuated chamber at temperature $T_{2} > T_{1}$ . The rate of growth of temperature is proportional to :

T_{2} - T_{1}

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{T_{2}}^{2} - {T_{1}}^{2}

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{T_{2}}^{3} - {T_{1}}^{3}

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{T_{2}}^{4} - {T_{1}}^{4}

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Solution

The correct option is D ${T_{2}}^{4} - {T_{1}}^{4}$
$P_{1} = A e σ {T_{1}}^{4}$ (rate of loss of radiant energy)
$P_{2} = A e σ {T_{2}}^{4}$ (rate of gain of radiant energy)
Net rate of gain of radiant energy
$(P_{2} - P_{1}) \propto ({T_{2}}^{4} - {T_{1}}^{4})$

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