Question

A spherical black body with a radius of 12 cm radiates 450 W power at 500 K. If the radius were halved and the temperature be doubled, the power radiated in watt would be:
A. 1800
B. 900
C. 3600
D. 850

Answer 1

Hint: In order to calculate the required power radiated by a body we will use the Stefan’s law for radiation which is expressed as:-
$P = \sigma Ae{T^4}$ , Where P is the power in watts (J/s) radiated by an object. A is the surface area in ${{\text{m}}^2}$, e is the emissivity of the object for black body.

Complete step-by-step answer:
To find the required power radiated by an object.
Formula used : $P = \sigma Ae{T^4}$…………….(i)
For the case of black body radiation we use the value of emissivity (e) =1.
Substituting the value of emissivity (e) =1 in the eqn (i).
We get:-
$P = \sigma A{T^4}$………………………….(ii)
According to question
We have given:-
${r_f} = \dfrac{1}{2}{r_i}$ and ${T_f} = 2{T_i}$
Where ${r_f}{\text{,}}{r_i}$ Final and initial radius of spherical - shaped black body and ${T_f}{\text{,}}{T_i}$ Final and initial temperature of spherical - shaped black body.
Using eqn (ii) for writing the expression for initial and final power radiated by the black body.
We get
${P_1} = \sigma {A_1}{T_1}^4$………………………(iii)
${P_2} = \sigma {A_2}{T_2}^4$……………………..(iv)
Dividing eqn (iv) by eqn (iii)
we get
\[
  \dfrac{{{P_2}}}{{{P_1}}} = \dfrac{{\sigma {A_2}{T_2}^4}}{{\sigma {A_1}{T_1}^4}} \\
   \Rightarrow \dfrac{{{P_2}}}{{{P_1}}} = \dfrac{{{A_2}{T_2}^4}}{{{A_1}{T_1}^4}} \\
   \Rightarrow \dfrac{{{P_2}}}{{{P_1}}} = \dfrac{{\pi {r_2}^2{T_2}^4}}{{\pi {r_1}^2{T_1}^4}} \\
\]
\[ \Rightarrow \dfrac{{{P_2}}}{{{P_1}}} = \dfrac{{{r_2}^2{T_2}^4}}{{{r_1}^2{T_1}^4}}\]…………………..(V)
Substituting the value of ${r_f} = \dfrac{1}{2}{r_i}$ and \[{T_f} = 2{T_i}{\text{ in eqn(v)}}\]
we get,
\[
   \Rightarrow \dfrac{{{P_2}}}{{{P_1}}} = {\left( {\dfrac{1}{2}} \right)^2} \times {\left( 2 \right)^4} \\
   \Rightarrow \dfrac{{{P_2}}}{{{P_1}}} = 4 \\
   \Rightarrow {P_2} = 4{P_1} \\
\]
   Since \[{P_1} = 450{\text{W}}\]….Given
\[
   \Rightarrow {P_2} = 4 \times 450{\text{W}} \\
   \Rightarrow {P_2} = 1800{\text{W}} \\
\]
Hence, Option (A) is the correct answer.

Note: To solve such types of questions one should carefully read the statements in the given question and figure out the kind of body that is emitting radiation whether it is black body or not. If black body is not mentioned in the question then we do not use the expression $P = \sigma A{T^4}$. We will use $P = \sigma Ae{T^4}$ where ,“e” lies in between 0 and 1.