The power radiated by a black body is P and it radiates maximum energy around the wavelength $\lambda_0$.…

The power radiated by a black body is P and it radiates maximum energy around the wavelength $\lambda_0$. Now the temperature of the black body is changed so that it radiates maximum energy around wavelength $\left(\frac{\lambda_0}{2}\right)$. The power radiated by it will now increase by a factor of
  1. 2
  2. 8
  3. 16
  4. 32

Solution

$\begin{array}{ll} & \text { According to Wien’s law, } \\ & \lambda_{\mathrm{m}} \mathrm{~T}=\text { constant }. \\ & \lambda_{m_1} T_1=\lambda_{m_2} T_2 \\ \therefore \quad & T_2=\frac{\lambda_{m_1}}{\lambda_{m_2}} T_1=\frac{\lambda_0}{\left(\frac{\lambda_0}{2}\right)} \times T_1=2 T_1...(i) \end{array}$
From Stefan-Boltzmann law, $\mathrm{P} \propto \mathrm{T}^4$ $\therefore \quad \frac{P_2}{P_r}=\left(\frac{T_2}{T_1}\right)^4$. $\therefore \quad \frac{P_2}{P_1}=\left(\frac{2 T_1}{T_1}\right)^4=16$

Asked in: MHT CET 2024 (16 May Shift 1)

Practice more Kinetic Theory of Gases and Radiation questions on Aicharya