Question

Radiation from a black body at the thermodynamic temperature $T_{1}$ is measured by a small detector at distance $d_{1}$ from it. When the temperature is increased to $T,$ and the distance to $d_{2}$, the power received by the detector is unchanged. What is the ratio $d_{2} / d_{1} ?$

• A. $\frac{T_{2}}{T_{1}}$
• B. $\left(\frac{T_{2}}{T_{1}}\right)^{2}$
• C. $\left(\frac{T_{1}}{T_{2}}\right)^{2}$
• D. $\left(\frac{T_{2}}{T_{1}}\right)^{4}$

Answer 1

Answer: (B)

Power received $\propto\left(\frac{E}{d^{2}}\right) \propto \frac{T^{4}}{d^{2}}$
Here $\frac{T_{2}^{4}}{d_{2}^{2}}=\frac{T_{1}^{4}}{d_{1}^{2}}$
$\Rightarrow \left(\frac{d_{2}}{d_{1}}\right)^{2}=\left(\frac{T_{2}}{T_{1}}\right)^{4}$ or $\frac{d_{2}}{d_{1}}=\left(\frac{T_{2}}{T_{1}}\right)^{2}$