Question

A black body (heat capacity = C) at absolute temperature T is kept in a surrounding of absolute temperature T/2. Its rate of cooling is :

($\sigma$ is stefan's constant)

• A. $\frac{\sigma A}{C}{\left(\frac{T}{2}\right)}^4$
• B. $\frac{\sigma A{T}^4}{C2}$
• C. $\frac{15\sigma A{T}^4}{C2}$
• D. $\frac{15\sigma A}{C}{\left(\frac{T}{2}\right)}^4$

Answer 1

Answer: (D)

$\frac{15\sigma A}{C}{\left(\frac{T}{2}\right)}^4$

Rate at which heat is radiated as given by Stefan's Law:

$P=\sigma A{T}^4$

Rate of cooling is $\frac{dQ}{dt}=ms\frac{dT}{dt}$
Equating both,

$ms\frac{dT}{dt}=\sigma A(T^4-(\frac{T}{2}{)}^4)$ where m is the mass and s is the specific heat capacity.

Heat capacity is $ms$.

Given, $ms=C$
Thus,

$C\frac{dT}{dt}=\sigma A\times 15\times (\frac{T}{2}{)}^4$

$\frac{dT}{dt}=\frac{15\sigma A}{C}\times (\frac{T}{2}{)}^4$