Question

If the temperature of hot body is raised by $5\% $ then the heat energy radiated would increase by:
A. $18\% $
B. $24\% $
C. $16\% $
D. $21\% $

Answer 1

Hint: Here we have to use the relation between energy and temperature to find the answer which is given by the Stefan-Boltzmann law. The Stefan-Boltzmann law states that the cumulative energy radiated per unit surface area of a black body over all wavelengths per unit time (also known as the radiant emittance of the black body) is directly proportional to the thermodynamic temperature T of the black body's fourth power.
Since, the energy is raised by $5\% $ so, we have to multiply $5\% $ to the temperature.

Complete step by step answer:
The relation between energy and temperature is given as:
$E + \dfrac{{dE}}
{E} = \dfrac{{{{\left[ {T + \dfrac{{5T}}
{{100}}} \right]}^4}}}
{{{T^4}}}$

So after taking energy and temperature as $1$
 we get:
$
 1 + \dfrac{{dE}}
{E} = \dfrac{{{{\left[ {1 + \dfrac{{5T}}
{{100}}} \right]}^4}}}
{{{1^4}}} \\
\Rightarrow 1 + \dfrac{{dE}}
{E} = {\left[ {1 + 1.05} \right]^4} \\
\Rightarrow \dfrac{{dE}}
{E} = 1.2155 - 1 = 0.2155 \\
\Rightarrow \dfrac{{dE}}
{E} = 21.55\% \\
$

So, the correct answer is “Option D”.

Additional Information:
- The amount of thermal radiation radiated per unit time depends on the nature, area and temperature of the emitting surface. At higher temperatures, the pace is faster. When the body radiates more than it absorbs, the temperature decreases. When the body radiates less than it consumes, the temperature increases.
- As the heat flows through the object, its form changes. Generally, as warm, the substance expands. This material property is called thermal expansion. There may be a difference in the area , volume, and shape of the material.
- In thermodynamics, thermal and thermal principles are closely connected to precise descriptions. Heat, thermal energy is moved from the hotter system to the cooler system in contact. Temperature is a calculation of the total kinetic energy of the atoms or molecules within the structure.
- Temperature measurement is the resulting effect of the interaction of kinetic energy between the device and the molecules or atoms within the measuring instrument (i.e. a thermometer using either mercury-atoms or alcohol-molecules) or an increase in thermal energy from the baseline using the infrared spectrum. The calculation of temperature is also the measurement of the total kinetic energy within the device.

Note:
Here we have to see the amount of percentage as how much energy is radiated. Also, here we have taken the value of energy and temperature as one. If the value had been provided in the question, then the answer would have been different. Here $dE$ is taken as there is change in energy.
If heat is added into the environment, the molecules consume thermal energy and increase their kinetic energy.