The black discs $\mathrm{x}$, $\mathrm{y}$, and $\mathrm{z}$ have radii of $1 \, \text{m}$, $2 \, \text{m}$, and $3 \, \text{m}$, respectively. The wavelengths corresponding to maximum intensity are $200 \, \text{nm}$, $300 \, \text{nm}$, and $400 \, \text{nm}$, respectively. Here, we explore the relationship between the emissive power $E_x$, $E_y$, and $E_z$ for these discs.

Emissive Power

The formula for emissive power is given by:

$ E = \sigma A T^4 \quad \Rightarrow \quad E \propto A T^4 $

Where:

$A$ is the area of the disc.

$T$ is the absolute temperature.

$\sigma$ is the Stefan-Boltzmann constant.

Area Calculation

The area $A$ of a disc is given by $A = \pi R^2$. Therefore:

$ A \propto R^2 $

Given:

$R_1 = 1 \, \text{m}$

$R_2 = 2 \, \text{m}$

$R_3 = 3 \, \text{m}$

The areas are proportional as follows:

$ A_x : A_y : A_z \quad \text{proportional to} \quad 1 : 4 : 9 \quad \text{... (i)} $

Temperature Analysis Using Wien's Displacement Law

By Wien's displacement law, we have:

$ \lambda_{\max} \, T = \text{constant} \quad \Rightarrow \quad T \propto \frac{1}{\lambda} $

Given maximum wavelengths:

$\lambda_{\max 1} = 200 \, \text{nm}$

$\lambda_{\max 2} = 300 \, \text{nm}$

$\lambda_{\max 3} = 400 \, \text{nm}$

The wavelengths are proportional as follows:

$ \lambda_x : \lambda_y : \lambda_z \quad \text{proportional to} \quad 2 : 3 : 4 $

Thus the temperatures are:

$ \frac{1}{T_x} : \frac{1}{T_y} : \frac{1}{T_z} \quad \text{proportional to} \quad 2 : 3 : 4 $

Therefore, the temperatures are proportional as:

$ T_x : T_y : T_z \quad \text{proportional to} \quad \frac{1}{2} : \frac{1}{3} : \frac{1}{4} $

Simplified, this gives:

$ T_x : T_y : T_z \quad \text{proportional to} \quad 6 : 4 : 3 \quad \text{... (ii)} $

Comparison of Emissive Powers

We compare the product $A T^4$ for the discs:

For disc $\mathrm{x}$: $A_x T_x^4 = 1 \times (6)^4 = 1296$

For disc $\mathrm{y}$: $A_y T_y^4 = 4 \times (4)^4 = 1024$

For disc $\mathrm{z}$: $A_z T_z^4 = 9 \times (3)^4 = 729$

Thus, the relationship between the emissive powers is:

$ E_x > E_y > E_z $

Heat And Thermodynamics

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