Question

Power emitted by a black body at temperature $50\;^\circ {\rm{C}}$ is P. Now, temperature is doubled i.e. the temperature of the blackbody becomes $100\;^\circ {\rm{C}}$. Now power emitted is:
(a). greater than P but less than $16P$
(b). greater than $16P$
(c). $P$
(d). $16P$

Answer 1

Hint: Here we are using Stefan’s Boltzmann law. The power emitted depends on the temperature at which the body is. In this law temperature is expressed in Kelvins.

Complete step by step answer:
Given the temperature of black body in the first considered case is $50\;^\circ {\rm{C}}$. Let $P$ be the power emitted by the body at this temperature. The second case is when the temperature was doubled. That is when the temperature is $100\;^\circ {\rm{C}}$. Let ${P_2}$ be the power emitted by the body at this temperature.
As we know, according to Stefan’s Boltzmann law, the power emitted by a black body is directly proportional to the fourth power of temperature. The law can further be expressed as,
$P \propto {T^4}$
So in this case, the above expression can take the form as,
$\dfrac{{{P_2}}}{P} = \dfrac{{{T_2}^4}}{{{T_1}^4}}$
Here the temperatures are expressed in Celsius.
But in Stefan’ expression the temperatures are to be expressed in Kelvin. So the given temperatures are to be converted to Kelvins by adding $273$. Thus we have,
$
{T_2} = 100\;^\circ {\rm{C}}\\
\Rightarrow{T_2} = 273 + 100\\
\Rightarrow{T_2} = 373\;{\rm{K}}
$
$
\Rightarrow{T_1} = 50\;^\circ {\rm{C}}\\
\Rightarrow{T_1} = 273 + 50\\
\Rightarrow{T_1} = 323\;{\rm{K}}
$
Thus the above expressed relation $\dfrac{{{P_2}}}{P} = \dfrac{{{T_2}^4}}{{{T_1}^4}}$ now becomes,
$
\dfrac{{{P_2}}}{P} = {\left( {\dfrac{{373}}{{323}}} \right)^4}\\
\Rightarrow\dfrac{{{P_2}}}{P} = {2^4}\\
\therefore{P_2} = {2^4}P
$
Therefore value of power emitted in case two is greater than P but less than $16$. So the correct option is: (a) greater than P but less than $16P$.

Note:The temperatures are to be converted into Kelvins and the unit of power is calculated in Watts. Stefan's Boltzmann law, also called Stefan’s law is only applicable to black bodies. When the Stefan’ law is related to power emitted the Wien’s displacement law deals with the thermal energy of a body.