Question

A blackened metallic foil kept at a distance $d$ from a spherical heater absorbs power $P$ . What will be the power absorbed by it if temperature of heater and distance are both doubled?

• A. $8P$
• B. $4P$
• C. $2P$
• D. $P$

Answer 1

Answer: (B)

Let P be the emissive power, T be the absolute temperature of body and $T_{0}$ be the temperature of surroundings.
According to Stefan's-Boltzmann's law, emissive power is
$P \propto \left(T^{4} - T_{0}^{4}\right)$
$\Rightarrow P \propto T^{4}\left(\because T_{0} < < T\right)$ .
Also Power of the radiation is inversely proportional to square of distance
$P \propto \frac{1}{d^{2}}$
Combining both, we get
$\Rightarrow P \propto \frac{T^{4}}{d^{2}}\Rightarrow \frac{P_{1}}{P_{2}}=\left(\frac{T_{1}}{T_{2}}\right)^{4}\times \left(\frac{d_{2}}{d_{1}}\right)^{2}$
$\Rightarrow \frac{P}{P_{2}}=\left(\frac{T}{2 T}\right)^{4}\times \left(\frac{2 d}{d}\right)^{2}=\frac{1}{4}\Rightarrow P_{2}=4P$