Question 856: Thermodynamics & KTG

Question

Two spherical black bodies of radii '$$r_1$$' and '$$r_2$$' at temperature '$$\mathrm{T}_1$$' and '$$\mathrm{T}_2$$' respectively radiate power in the ratio $$1: 2$$ Then $$r_1: r_2$$ is

Accepted Answer

Power Radiated by black body, $$\mathrm{P}=\sigma \mathrm{AT}^4$$
$$	herefore \quad$$ For first black body:
$$\mathrm{P}_1=\sigma 4 \pi \mathrm{r}_1^2 \mathrm{~T}_1^4$$
$$	herefore \quad$$ For second black body:
$$\mathrm{P}_2=\sigma 4 \pi \mathrm{r}_2{ }^2 \mathrm{~T}_2{ }^4$$
$$	herefore \quad$$ The ratio will be:
$$\begin{aligned}
& \frac{\mathrm{P}_1}{\mathrm{P}_2}=\frac{\sigma 4 \pi \mathrm{r}_1^2 \mathrm{~T}_1^4}{\sigma 4 \pi \mathrm{r}_2^2 \mathrm{~T}_2^4} \
& \frac{1}{2}=\frac{\mathrm{r}_1^2 \mathrm{~T}_1^4}{\mathrm{r}_2^2 \mathrm{~T}_2^4} \
& \frac{\mathrm{r}_1^2}{\mathrm{r}_2^2}=\frac{1}{2} \frac{\mathrm{T}_2^4}{\mathrm{~T}_1^4} \
& \frac{\mathrm{r}_1}{\mathrm{r}_2}=\frac{1}{\sqrt{2}}\left(\frac{\mathrm{T}_2}{\mathrm{~T}_1}ight)^2
\end{aligned}$$

Answer

$$\frac{1}{2}\left(\frac{\mathrm{T}_2}{\mathrm{~T}_1}\right)^4$$

Answer

$$\frac{1}{\sqrt{2}}\left(\frac{\mathrm{T}_2}{\mathrm{~T}_1}\right)^2$$

Answer

$$2\left(\frac{\mathrm{T}_1}{\mathrm{~T}_2}\right)^4$$

Answer

$$2\left(\frac{T_1}{T_2}\right)^2$$

Thermodynamics & KTG

Two spherical black bodies of radii '$r_1' and 'r_2' at temperature '\mathrm{T}_1' and '\mathrm{T}_2' respectively radiate power in the ratio 1: 2 Then r_1: r_2$ is

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