JEE Physics Practice Question

A current carrying wire heats a metal rod. The wire provides a constant power $(P)$ to the rod. The metal rod is enclosed in an insulated container. It is observed that the temperature $(T)$ in the metal rod changes with time $(t)$ as \[ T(t)=T_{0}\left(1+\beta t^{\frac{1}{4}}\right) \] where $\beta$ is a constant with appropriate dimension while $T_{0}$ is a constant with dimension of temperature. The heat capacity of the metal is,

1. A.$\frac{4 P\left(T(t)-T_{0}\right)^{3}}{\beta^{4} T_{0}^{4}}$
2. B.$\frac{4 P\left(T(t)-T_{0}\right)^{4}}{\beta^{4} T_{0}^{5}}$
3. C.$\frac{4 P\left(T(t)-T_{0}\right)^{2}}{\beta^{4} T_{0}^{3}}$
4. D.$\frac{4 P\left(T(t)-T_{0}\right)}{\beta^{4} T_{0}^{2}}$