A black coloured solid sphere of radius \(R\) and mass \(M\) is inside a cavity with vacuum inside. The walls of the cavity are maintained at temperature \(T_0\). The initial temperature of the sphere is \(3T_0\). If the specific heat of the material of the sphere varies as \(\alpha T^3\) per unit mass with the temperature \(T\) of the sphere, where \(\alpha \) is a constant, then the time taken for the sphere to cool down to temperature \(2T_0\) will be ( \(\sigma \) is Stefan Boltzmann constant)

1. A \(\frac{{M\alpha }}{{4\pi {R^2}\sigma }}\,\ln \left( {\frac{3}{2}} \right)\)
2. B \(\frac{{M\alpha }}{{4\pi {R^2}\sigma }}\,\ln \left( {\frac{16}{3}} \right)\)
3. C \(\frac{{M\alpha }}{{16\pi {R^2}\sigma }}\,\ln \left( {\frac{16}{3}} \right)\)
4. D \(\frac{{M\alpha }}{{16\pi {R^2}\sigma }}\,\ln \left( {\frac{3}{2}} \right)\)