The Wien's displacement law (sometimes referred to as the Vienna transition law) is crucial in understanding the relationship between the temperature of a black body and the wavelength at which its radiation intensity is at its peak. Here's a structured proof based on Planck's law, which describes the intensity of radiation emitted by a black body at a given temperature.

Step 1: Planck's Law for Black Body Radiation

The spectral radiance $I(\lambda, T)$, which gives the power emitted per unit area per unit wavelength by a black body at temperature $T$, is given by Planck's law:

$I(\lambda, T) = \frac{2hc^2}{\lambda^5} \cdot \frac{1}{e^{\frac{hc}{\lambda kT}} - 1}$

where:

• $I(\lambda, T)$ is the intensity as a function of wavelength $\lambda$ and temperature $T$,
• $h$ is Planck's constant,
• $c$ is the speed of light,
• $k$ is Boltzmann's constant,
• $\lambda$ is the wavelength,
• $T$ is the temperature.

Step 2: Maximizing the Radiation Intensity

To find the wavelength $\lambda_{\text{max}}$ at which the radiation intensity is greatest, we need to maximize the function $I(\lambda, T)$ with respect to $\lambda$.

That is, we need to solve:

$\frac{dI(\lambda, T)}{d\lambda} = 0$

This process requires differentiating Planck's law with respect to $\lambda$. The derivative involves applying the product and chain rules, making this a fairly complex expression.

Step 3: Simplifying the Equation

To simplify the calculation, we can introduce a substitution: $x = \frac{hc}{\lambda kT}$

This substitution simplifies the expression for $I(\lambda, T)$ and allows us to rewrite Planck's law in terms of $x$ and $\lambda$. After differentiating and simplifying, we get the following transcendental equation:

$5(1 - e^{-x}) - x = 0$

Step 4: Solving the Equation Numerically

The above equation does not have an analytical solution, but it can be solved numerically. The solution for $x$ is approximately:

$x \approx 4.9651$

Step 5: Apply the Substitution Back

Using the substitution $x = \frac{hc}{\lambda_{\text{max}} kT}$, we can now solve for $\lambda_{\text{max}}$:

$\lambda_{\text{max}} = \frac{hc}{4.9651 kT}$

Step 6: Wien's Displacement Law

Finally, this result gives us Wien's displacement law:

$\lambda_{\text{max}} T = \frac{hc}{4.9651 k} \approx 2.898 \times 10^{-3} \, \text{m} \cdot \text{K}$

This shows that the wavelength $\lambda_{\text{max}}$ at which the intensity of the radiation is greatest is inversely proportional to the temperature $T$ of the black body.

Conclusion

The intensity of radiation is greatest at the wavelength $\lambda_{\text{max}}$, which is inversely proportional to the temperature of the black body, as described by Wien's displacement law. This law is a result of differentiating and maximizing Planck's radiation law.

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Do you need further details or have any questions? Here are five related questions that extend this information:

1. How does the intensity of radiation change with temperature at a fixed wavelength?
2. Can Wien's displacement law be applied to non-black bodies?
3. What are the physical implications of the peak wavelength shifting with temperature?
4. How does Planck's law relate to the Stefan-Boltzmann law?
5. What are the limits of applicability for Wien's displacement law?

Tip: The Wien’s displacement law helps in determining the temperature of stars by observing their peak wavelength of emitted radiation.