Wiens's Law can be derived from Planck's law by finding the wavelength at which the blackbody radiation curve reaches its maximum. Here's the step-by-step derivation:

### 1. **Planck’s Law of Blackbody Radiation:**

Planck’s law gives the spectral radiance \( I(\lambda, T) \) of a blackbody at a given temperature \( T \) as a function of wavelength \( \lambda \):

\[
I(\lambda, T) = \frac{2hc^2}{\lambda^5} \cdot \frac{1}{e^{\frac{hc}{\lambda k_B T}} - 1}
\]

Where:
- \( I(\lambda, T) \) is the spectral radiance,
- \( h \) is Planck’s constant,
- \( c \) is the speed of light,
- \( \lambda \) is the wavelength,
- \( k_B \) is Boltzmann’s constant,
- \( T \) is the temperature.

### 2. **Maximizing the Spectral Radiance:**

To find Wiens's law, we need to find the wavelength \( \lambda_{\text{max}} \) at which the spectral radiance \( I(\lambda, T) \) is maximized. This involves differentiating \( I(\lambda, T) \) with respect to \( \lambda \) and setting the result to zero:

\[
\frac{d}{d\lambda} \left[ \frac{2hc^2}{\lambda^5} \cdot \frac{1}{e^{\frac{hc}{\lambda k_B T}} - 1} \right] = 0
\]

This is a complex expression, but after performing the differentiation and simplifying (using numerical methods or approximation), the result gives the condition for maximum radiation:

\[
\lambda_{\text{max}} T = \frac{b}{T}
\]

where \( b \) is a constant known as **Wien's displacement constant**, and its value is approximately:

\[
b \approx 2.897 \times 10^{-3} \, \text{m} \cdot \text{K}
\]

Thus, **Wien's Law** is:

\[
\lambda_{\text{max}} = \frac{b}{T}
\]

### Interpretation:
- \( \lambda_{\text{max}} \) is the wavelength at which the blackbody emits the most radiation.
- As temperature \( T \) increases, the peak wavelength \( \lambda_{\text{max}} \) shifts to shorter wavelengths.

### Summary:
Wiens's law can be seen as the result of maximizing Planck's law for blackbody radiation. It shows the relationship between the temperature of a blackbody and the wavelength at which it emits the most radiation.

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**Related questions:**
1. What is the physical significance of Wien's law in astrophysics?
2. How does Wien's law relate to the color of stars?
3. Can you explain the derivation of Planck's law from first principles?
4. What is Stefan-Boltzmann's law and how does it relate to Planck's law?
5. How is the CMB (Cosmic Microwave Background) radiation analyzed using Wien’s law?

**Tip:** In the context of blackbody radiation, **as temperature increases, the peak wavelength of emission decreases**, leading to a shift toward higher energy radiation like visible light or ultraviolet.

Explore the detailed derivation of Wien's Law from Planck's Law of blackbody radiation. This guide explains key thermodynamic principles, such as spectral radiance and differentiation, and uses critical formulas like Planck's law and Wien's displacement law. Ideal for advanced university physics students.

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