Question

Temperature of a piece of metal is increased from $27^\circ \,C$ to $327^\circ \,C$. the rate of heat of radiation by the metal will become
A. Double
B. Four times
C. Eight times
D. Sixteen times

Answer 1

Hint: It is given in the question that a metal is radiating heat when the temperature will increase from $27^\circ \,C$ to $327^\circ \,C$ . Therefore, we will use Stefan Boltzmann law to find the rate of heat radiated by a metal. Stefan-Boltzmann law states that the total energy emitted by a black body per unit surface area is directly proportional to the fourth power of the temperature of a black body.

Complete step by step answer:
It is given in the question that the temperature of the metal is increased from $27^\circ \,C$ to $327^\circ \,C$. Now, changing the temperature from Celsius into kelvin.
Therefore, the initial temperature $ = \,27^\circ C\, = 27 + 273\, = 300\,K$
Also, final temperature $ = \,327^\circ C\, = \,327 + 273 = \,600K$

Now, the heat radiated by the metal can be calculated by using Stefan Boltzmann law. Stefan-Boltzmann law states that the total energy emitted by a black body per unit surface area is directly proportional to the fourth power of the temperature of a black body is given by,
$\varepsilon = \sigma {T^4}$
Where, $\sigma $ is the constant of proportionality.
Now, $\varepsilon = \dfrac{P}{A}$
Putting this value in the above equation, we get
$P = A\sigma {T^4}$
From here, we can say that, $P \propto \,{T^4}$

Now, taking the ratio of the heat radiated by the metal during the initial and final temperatures, we get
$\dfrac{{{P_1}}}{{{P_2}}} = {\left( {\dfrac{{{T_1}}}{{{T_2}}}} \right)^4}$
Now, putting the values of temperatures in the above equation, we get
$\dfrac{{{P_1}}}{{{P_2}}} = {\left( {\dfrac{{300}}{{600}}} \right)^4}$
$ \Rightarrow \,\dfrac{{{P_1}}}{{{P_2}}} = {\left( {\dfrac{1}{2}} \right)^4}$
$ \Rightarrow \,\dfrac{{{P_1}}}{{{P_2}}} = \dfrac{1}{{16}}$
$ \therefore \,{P_2} = 16{P_1}$

Therefore, the rate of heat radiation by the metal will become sixteen times.Hence, option D is the correct option.

Note:Here, remember to change the temperature from Celsius to kelvin. We change Celsius to kelvin because when we will talk about $0K$ it represents zero kinetic energy or temperature. But the Celsius or Fahrenheit scale does not begin with zero. We use kelvin as a unit of temperature because it is directly related to the kinetic energy and the temperature.