The black discs $x, y$ and $z$ have radii $1 \mathrm{~m}, 2 \mathrm{~m}$ and 3 m respectively. The…

The black discs $x, y$ and $z$ have radii $1 \mathrm{~m}, 2 \mathrm{~m}$ and 3 m respectively. The wavelengths corresponding to maximum intensity are $200 \mathrm{~nm}, 300 \mathrm{~nm}$ and 400 nm respectively. The relation between emissive power $\mathrm{E}_{\mathrm{x}}, \mathrm{E}_{\mathrm{y}}$ and $\mathrm{E}_{\mathrm{z}}$ is
  1. $\mathrm{E}_{\mathrm{x}}\gt\mathrm{E}_{\mathrm{y}}\gt\mathrm{E}_{\mathrm{z}}$
  2. $\mathrm{E}_{\mathrm{x}} \lt \mathrm{E}_{\mathrm{y}} \lt \mathrm{E}_{\mathrm{z}}$.
  3. $\quad \mathrm{E}_{\mathrm{x}}=\mathrm{E}_{\mathrm{y}}=\mathrm{E}_{\mathrm{z}}$
  4. $\mathrm{E}_{\mathrm{y}}\gt\mathrm{E}_{\mathrm{x}} \lt \mathrm{E}_z$

Solution

$\begin{array}{ll} \text { Emissive power } E=\sigma A T^4 & \Rightarrow E \propto A T^4 \\ A=\pi R^2 & \Rightarrow A \propto R^2 \end{array}$
Given, $\mathrm{R}_1=1 \mathrm{~m}, \mathrm{R}_2=2 \mathrm{~m}, \mathrm{R}_3=3 \mathrm{~m}$ $\Rightarrow A_x: A_y: A_z:: 1: 4: 9$
By Wien's displacement law $\lambda_{\max } \mathrm{T}=\text { constant } \quad \Rightarrow \mathrm{T} \propto \frac{1}{\lambda}$
Given, $\begin{aligned} & \lambda_{\max 1}=200 \mathrm{~nm}, \lambda_{\max 2}=300 \mathrm{~nm}, \lambda_{\max 3}=400 \mathrm{~nm} \\ & \Rightarrow \lambda_x: \lambda_y: \lambda_z:: 2: 3: 4 \\ & \frac{1}{\mathrm{~T}_x}: \frac{1}{\mathrm{~T}_{\mathrm{y}}}: \frac{1}{\mathrm{~T}_{\mathrm{z}}}:: 2: 3: 4 \end{aligned}$ $\begin{aligned} \therefore \quad & T_x: T_y: T_z:: \frac{1}{2}: \frac{1}{3}: \frac{1}{4} \\ & \text { or } T_x: T_y: T_z:: 6: 4: 3...(ii) \end{aligned}$
Comparing the product $\mathrm{AT}^4$ for the 3 discs From (i) and (ii), we have, for disc $\mathrm{x}: \mathrm{A}_{\mathrm{x}} \mathrm{T}_{\mathrm{x}}^4=1 \times(6)^4=1296$ for disc $\mathrm{y}: \mathrm{A}_{\mathrm{y}} \mathrm{T}_{\mathrm{y}}^4=4 \times(4)^4=1024$ for disc $z: A_z T_z^4=9 \times(3)^4=729$ $\therefore \quad \mathrm{E}_{\mathrm{x}}\gt\mathrm{E}_{\mathrm{y}}\gt\mathrm{E}_{\mathrm{z}}$.

Asked in: MHT CET 2024 (10 May Shift 2)

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