Wien's displacement law and its transformation can be derived using principles from Planck's law of blackbody radiation. Here’s a step-by-step outline of how you can derive **Wien's displacement law** from Planck's radiation law:

### 1. **Planck’s Law:**
Planck’s law gives the spectral radiance \( I(\lambda, T) \) of a blackbody as a function of wavelength \( \lambda \) and temperature \( T \):
\[
I(\lambda, T) = \frac{2hc^2}{\lambda^5} \cdot \frac{1}{e^{\frac{hc}{\lambda k_B T}} - 1}
\]
where:
- \( h \) is Planck's constant,
- \( c \) is the speed of light,
- \( k_B \) is Boltzmann's constant,
- \( \lambda \) is the wavelength,
- \( T \) is the temperature.

### 2. **Find the Peak Wavelength:**
To get Wien’s displacement law, we need to find the wavelength \( \lambda_{max} \) at which the spectral radiance \( I(\lambda, T) \) is maximum for a given temperature \( T \). This requires differentiating Planck's law with respect to \( \lambda \) and setting the result equal to zero:
\[
\frac{d}{d\lambda} I(\lambda, T) = 0
\]

### 3. **Differentiate Planck's Law:**
This step involves using the quotient rule to differentiate the expression for \( I(\lambda, T) \). It results in a transcendental equation that cannot be solved directly in a simple form but is typically solved numerically.

After differentiating, the equation becomes:
\[
5 \left( 1 - e^{- \frac{hc}{\lambda k_B T}} \right) = \frac{hc}{\lambda k_B T} e^{- \frac{hc}{\lambda k_B T}}
\]

This equation can be solved numerically for the value of \( \lambda_{max} T \).

### 4. **Wien's Displacement Law:**
The result of solving the transcendental equation provides the constant that relates the peak wavelength \( \lambda_{max} \) to the temperature \( T \). This is Wien's displacement law:
\[
\lambda_{max} T = b
\]
where \( b \) is **Wien's displacement constant**, with a value of approximately \( 2.898 \times 10^{-3} \, \text{m} \cdot \text{K} \).

Thus, the peak wavelength of blackbody radiation is inversely proportional to the temperature.

### Summary:
1. Start from **Planck's law** of blackbody radiation.
2. Differentiate the spectral radiance \( I(\lambda, T) \) with respect to wavelength \( \lambda \).
3. Solve the resulting transcendental equation numerically.
4. The result is Wien's displacement law: \( \lambda_{max} T = b \), where \( b \approx 2.898 \times 10^{-3} \, \text{m} \cdot \text{K} \).

Would you like further clarification on any of the steps?

Here are 5 questions related to this:
1. How is Planck's law used to describe blackbody radiation?
2. What are the practical applications of Wien's displacement law in astrophysics?
3. How is Wien's displacement constant derived numerically?
4. What are the limitations of Wien's displacement law for extremely high or low temperatures?
5. How does Wien's displacement law connect with the Stefan-Boltzmann law?

**Tip:** Wien’s displacement law is useful in astrophysics to determine the temperature of stars by observing their peak emission wavelength!

Explore the step-by-step derivation of Wien's displacement law from Planck's radiation law. This process involves using spectral radiance, differentiation, and solving transcendental equations. Learn how to find the peak wavelength of blackbody radiation and understand Wien's constant. Suitable for undergraduate physics students.

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