Three black discs ' $\mathrm{x}^{\prime}$, $^{\prime} \mathrm{y}^{\prime},{ }^{\prime} \mathrm{z}^{\prime}$…

Three black discs ' $\mathrm{x}^{\prime}$, $^{\prime} \mathrm{y}^{\prime},{ }^{\prime} \mathrm{z}^{\prime}$ have radii $1 \mathrm{~m}, 2 \mathrm{~m}$ and $3 \mathrm{~m}$ respectively. The wavelengths corresponding to maximum intensity are 200,300 and $400 \mathrm{~nm}$ respectively. The relation between emissive powers 'Ex', 'Ey' and 'Ez' is
  1. $\mathrm{Ex}>\mathrm{Ey} < \mathrm{Ez}$
  2. $\mathrm{Ex} < \mathrm{Ey} < \mathrm{Ez}$
  3. $\mathrm{Ex}=\mathrm{Ey}=\mathrm{Ez}$
  4. $\mathrm{Ex}>\mathrm{Ey}>\mathrm{Ez}$

Solution

$\lambda_{\mathrm{m}} \mathrm{T}=\mathrm{k}$ $\mathrm{E}=\frac{\sigma \mathrm{AT}^{4}}{\mathrm{~A}} \Rightarrow \mathrm{E} \propto \mathrm{T}^{4} \quad \therefore \mathrm{E} \propto \frac{\mathrm{A}}{\lambda^{4}}$ $\mathrm{E}_{\mathrm{x}}: \mathrm{E}_{\mathrm{y}}: \mathrm{E}_{\mathrm{z}}=\frac{1}{\lambda_{\mathrm{x}}^{4}}: \frac{1}{\lambda_{\mathrm{y}}^{4}}: \frac{1}{\lambda_{\mathrm{z}}^{4}}=\frac{1}{(200)^{4}}: \frac{1}{(300)^{4}}: \frac{1}{(400)^{4}}$ $\mathrm{E}_{\mathrm{x}}>\mathrm{E}_{\mathrm{y}}>\mathrm{E}_{\mathrm{z}}$

Asked in: MHT CET 2020 (20 Oct Shift 1)

Practice more Kinetic Theory of Gases and Radiation questions on Aicharya