Prove that the intensity of the radiation is greatest according to the Vienna transition law

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.

Discover the proof of Wien's Displacement Law, showing how the intensity of black body radiation reaches its peak at a specific wavelength, inversely proportional to temperature. Explore key concepts such as Planck's Law, black body radiation, and the mathematical derivation using calculus. This detailed guide is perfect for undergraduate physics students.

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