Question

Two bodies A and B are kept in an evacuated chamber at${27^ \circ }C$. The temperature of A and B are ${327^ \circ }C{\text{ and }}{427^ \circ }C$ respectively. The ratio of rates of loss of heat from A and B will be:
(A). 0.23
(B). 0.52
(C). 1.52
(D). 2.52

Answer 1

Hint: To solve this question easily students need a proper understanding of Stefan- Boltzmann Law which states that the energy of thermal radiation emitted per unit time by a black body of surface area A is directly proportional to a universal constant known as Stefan constant, the surface area A and temperature T on absolute scale raised to power four.

Complete step by step answer:
Consider a body a body of emission e kept in thermal equilibrium in a room at temperature ${T_ \circ }$. The energy of radiation absorbed by it per unit time should be equal to the energy emitted by it per unit time. This is because the temperature remains constant. Thus, the energy of the radiation absorbed per unit time is:
$u = e\sigma AT_ \circ ^4$
Now, suppose the temperature of the body is changed to T but the room temperature remains${T_ \circ }$. The energy of the thermal radiation emitted by the body per unit time is:
$u = e\sigma A{T^4}$
The energy absorbed per unit time by the body is:
 ${u_ \circ } = e\sigma A{T_ \circ }^4$
Thus, the net loss of thermal energy per unit time is:
$\eqalign{
  & \Delta u = u - {u_ \circ } \cr
  & \Rightarrow \Delta u = e\sigma A\left( {{T^4} - T_ \circ ^4} \right) \cr
  & {\text{Additionally,}} \cr
  & e\sigma A\left( {{T^4} - T_ \circ ^4} \right) = \dfrac{{\Delta Q}}{{\Delta t}} \cr} $
So, Q is directly proportional to the temperature difference.
$\eqalign{
  & Q \propto \left( {{T^4} - T_ \circ ^4} \right) \cr
  & \Rightarrow \dfrac{{{Q_1}}}{{{Q_2}}} = \dfrac{{{T_1}^4 - T_ \circ ^4}}{{{T_2}^4 - T_ \circ ^4}} \cr} $
Given:
$\eqalign{
  & {T_0} = 27 + 273 = 300K \cr
  & {T_1} = 327 + 273 = 600K \cr
  & {T_2} = 427 + 273 = 700K \cr} $
Substituting values in the above equation, we get:
$\eqalign{
  & \dfrac{{Q{}_1}}{{{Q_2}}} = \dfrac{{{{600}^4} - {{300}^4}}}{{{{700}^4} - {{300}^4}}} \cr
  & = \dfrac{{{6^4} - {3^4}}}{{{7^4} - {3^4}}} \cr
  & = \dfrac{{{{36}^2} - {9^2}}}{{{{49}^2} - {9^2}}} \cr
  & = \dfrac{{243}}{{464}} \cr
  & = 0.52 \cr} $
Therefore, the correct option is B. , i.e. 0.52.

Note: A brief way to solve this same question is to describe Stefan- Boltzmann Law. Then take ratio of its two equations formed by the given values in the question i.e.,
$\eqalign{
  & \dfrac{{E{}_1}}{{{E_2}}} = \dfrac{{\sigma \left( {T_1^4 - T_ \circ ^4} \right)}}{{\sigma \left( {T_2^4 - T_ \circ ^4} \right)}} = \dfrac{{\left( {{{600}^4} - {{300}^4}} \right)}}{{\left( {{{700}^4} - {{300}^4}} \right)}} \cr
  & = 0.52 \cr} $