The wavelength of maximum emitted energy $\left(\lambda_m\right)$ of a body at $700 \mathrm{~K}$ is $4.08…

The wavelength of maximum emitted energy $\left(\lambda_m\right)$ of a body at $700 \mathrm{~K}$ is $4.08 \mu \mathrm{m}$. If the temperature of the body is raised to $1400 \mathrm{~K}$, then the value of $\lambda_m$ will be
  1. ) $1.02 \mu \mathrm{m}$
  2. $16.32 \mu \mathrm{m}$
  3. $8.16 \mu \mathrm{m}$
  4. $2.04 \mu \mathrm{m}$

Solution

Given, $ \begin{aligned} & \lambda_{m_1}=4.08 \mu \mathrm{m} \\ & T_1=700 \mathrm{~K} \\ & T_2=1400 \mathrm{~K} \\ & \lambda_{m_2}=\lambda_m=? \end{aligned} $ According to Wien's displacement law, $ \begin{aligned} & \lambda_m \propto \frac{1}{T} \Rightarrow \frac{\lambda_{m_2}}{\lambda_{m_1}}=\frac{T_1}{T_2} \\ \Rightarrow \quad & \lambda_{m_2}=\frac{T_1 \lambda_{m_1}}{T_2}=\frac{700 \times 4.08}{1400}=2.04 \mu \mathrm{m} \\ \therefore \quad & \lambda_m=\lambda_{m_2}=2.04 \mu \mathrm{m} \end{aligned} $

Asked in: AP EAMCET 2020 (22 Sep Shift 2)

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