# The temperature of an isolated black body falls from T1 to T2 in time ‘t’. Let ‘c’ be a constant, then……

Q: The temperature of an isolated black body falls from T1 to T2 in time ‘t’. Let ‘c’ be a constant, then……

(a) $\displaystyle t = c (\frac{1}{T_2}-\frac{1}{T_1} )$

(b) $\displaystyle t = c (\frac{1}{{T_2}^2}-\frac{1}{{T_1}^2} )$

(c) $\displaystyle t = c (\frac{1}{{T_2}^3}-\frac{1}{{T_1}^3} )$

(d) $\displaystyle t = c (\frac{1}{{T_2}^4}-\frac{1}{{T_1}^4} )$

Ans: (c)

Sol: $\displaystyle ms\frac{dT}{dt} = -\sigma A(T^4 – 0)$

$\displaystyle ms\frac{dT}{dt} = -\sigma A T^4$

$\displaystyle ms\int_{T_1}^{T_2} \frac{dT}{T^4} = -\sigma A \int_{0}^{t}dt$

$\displaystyle \frac{ms}{3} (\frac{1}{T_2^3 – \frac{1}{T_1^3}}) = \sigma A t$

$\displaystyle t = \frac{ms}{3\sigma A} (\frac{1}{T_2^3 – \frac{1}{T_1^3}})$

$\displaystyle t = c (\frac{1}{T_2^3 – \frac{1}{T_1^3}})$