Question

Two copper spheres of radii $6cm$ and $12cm$ respectively are suspended in an evacuated enclosure. Each of them are at temperature $15^{\circ}C$ above the surrounding. The ratio of loss of heat is:

& A.2:1 \\\ & B.1:4 \\\ & C.1:8 \\\ & D.8:1 \\\ \end{aligned}$$

Answer 1

To calculate the ratio of loss as heat, we can use the Stefan-Boltzmann law. Which gives the relationship between the radiation and the degree of the power. Since all the necessary quantities are given, we can substitute and calculate the ratio of heat emitted.

Formula used:
$E\propto T^{4}$ and $E=\sigma A{\Delta T}=4\pi r^{2} \sigma \Delta T$

Complete answer:
Here, we have two spheres of the same material i.e. copper. Given that their radii are $r_{1}=6cm$ and the other is $r_{2}=12cm$.

We know that radiation is the degree of how much power is emitter, reflected or transmitted by anybody.
Similarly we know that, according to Stefan-Boltzmann law, which states that, the total radiation of heat emitted from a surface is proportional to the fourth power of its absolute temperature.
$E\propto T^{4}$
$E=\sigma T^{4}$, where $\sigma$ is the constant of proportionality called the Stefan-Boltzmann constant also, $\sigma=5.670\times 10^{-8} W/mK^{4}$
Also given as $E=\sigma A{\Delta T}=4\pi r^{2} \sigma \Delta T$
Then the ratio is given as $\dfrac{E_{1}}{E_{2}}=\dfrac{4\pi r^{2}_{1} \sigma \Delta T}{4\pi r^{2}_{2} \sigma \Delta T}$
Since the $\Delta T$ for both the spheres are $15^{\circ} C$, we get, the ratio of heat loss as $\dfrac{E_{1}}{E_{2}}=\dfrac{r^{2}_{1}}{r^{2}_{2}}$
Substituting the values for $r_{1}=6cm$ and $r_{2}=12cm$, we get
$\dfrac{E_{1}}{E_{2}}=\dfrac{6^{2}}{12^{2}}=\left(\dfrac{1}{2}\right)^{2}=\dfrac{1}{4}$

So, the correct answer is “Option B”.

Additional Information:
The Stefan-Boltzmann law talks about the heat emitted due to a black body in terms of temperature of the body. According to it, the total energy radiated per unit area of the black body across the wavelengths per unit time is proportional to temperature raised to the fourth power. It uses the basics of thermodynamics and Planck’s law.
This law is used to determine the temperature of the sun’s surface, radiations emitted by the stars and effective temperature of the earth.

Note:
The Stefan-Boltzmann law gives the radiation of heat produced due to a mass of an object. Also note that, since both the bodies are at same temperature, $E\propto A$, where $A$ is the area of the body. This law is used to determine the temperature of the sun’s surface, radiation emitted by the stars and effective temperature of the earth.