Question

Two perfectly black spheres $A$ and $B$ having radii $8\,cm$ and $2\,cm$ are maintained at temperatures $127^{\circ}C$ and $527^{\circ}C$ respectively. The ratio of the energy radiated by $A$ to that by $B$ is

• A. 1:02
• B. 1:01
• C. 2:01
• D. 1:04

Answer 1

Answer: (B)

1:01

Answer 2

Total energy emitted per second by a unit area of a body is proportional to the fourth power of its absolute temperature.
i.e. $E=\sigma T^{4}$
where, $\sigma$ is a universal constant called Stefan Boltzmann constant.
Given, $T_{1} =127^{\circ} C$
$=127^{\circ}+273=400\, K$
$T_{2} =527^{\circ} C =800\, K$
and $r_{1}=8\, cm \Rightarrow r_{2}=2\, cm$
We know that total energy,
$E=\sigma A T^{4}$
In the first condition,
$E_{1}=\sigma_{1} A_{1} T_{1}^{4}$ ...(i)
In the second condition,
$E_{2}=\sigma_{2} A_{2} T_{2}^{4}$ ...(ii)
On dividing E (i) by E (ii), we get
$\frac{E_{1}}{E_{2}} =\frac{A_{1} T_{1}^{4}}{A_{2} T_{2}} {\left[\therefore \sigma_{1}=\sigma_{2}\right]}$
$=\frac{8^{2}}{2^{2}} \times \frac{(400)^{4}}{(800)^{4}}$
$\Rightarrow \frac{E_{1}}{E_{2}} =4^{2} \times \frac{1}{2^{4}}$
$\Rightarrow \frac{E_{1}}{E_{2}} =\frac{1}{1}$ or $1: 1$