A current carrying wire heats a metal rod. The wire provides a constant power $\left(\mathrm{P}\right)$ to the rod. The metal rod is enclosed in an insulated container. It is observed that the temperature $\left(\mathrm{T}\right)$ in the metal rod changes with time $\left(\mathrm{t}\right)$ as:
$\mathrm{T}\left(\mathrm{t}\right)={\mathrm{T}}_0\left(1+\mathrm{\beta }{\mathrm{t}}^{1/4}\right)$
where $\mathrm{\beta }$ is a constant with appropriate dimension while ${\mathrm{T}}_0$ is a constant with dimension of temperature. The heat capacity of the metal is:

• A

  $\frac{4\mathrm{P}\left(\mathrm{T}\left(\mathrm{t}\right)-{\mathrm{T}}_0\right)}{{\mathrm{\beta }}^4{\mathrm{T}}_0^2}$
• B

  $\frac{4\mathrm{P}{\left(\mathrm{T}\left(\mathrm{t}\right)-{\mathrm{T}}_0\right)}^2}{{\mathrm{\beta }}^4{\mathrm{T}}_0^3}$
• C

  $\frac{4\mathrm{P}{\left(\mathrm{T}\left(\mathrm{t}\right)-{\mathrm{T}}_0\right)}^4}{{\mathrm{\beta }}^4{\mathrm{T}}_0^5}$
• Correct answer
  D

  $\frac{4\mathrm{P}{\left(\mathrm{T}\left(\mathrm{t}\right)-{\mathrm{T}}_0\right)}^3}{{\mathrm{\beta }}^4{\mathrm{T}}_0^4}$

Hence, the correct answer is option: 'D'.