wins law from planks law. show steps in detail

To derive **Wien's Displacement Law** from **Planck's Law**, we need to connect the peak wavelength of blackbody radiation to the temperature of the blackbody.

### **Step-by-Step Derivation:**

### 1. **Start with Planck's Law:**
Planck's law gives the spectral radiance \( I(\lambda, T) \) of a blackbody as a function of wavelength \( \lambda \) and temperature \( T \):

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

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 of the blackbody.

### 2. **Find the Peak of Spectral Radiance:**
To find the wavelength \( \lambda_{\text{max}} \) at which the radiance is maximum for a given temperature, we need to take the derivative of \( I(\lambda, T) \) with respect to \( \lambda \), set it equal to zero, and solve for \( \lambda_{\text{max}} \). That is:

\[
\frac{d}{d\lambda} \left( I(\lambda, T) \right) = 0
\]

### 3. **Simplify Planck's Law:**

For ease, let’s define the following substitution:

\[
x = \frac{hc}{\lambda k_B T}
\]

This substitution simplifies the Planck function to:

\[
I(\lambda, T) = \frac{2\pi k_B^5 T^5}{h^4 c^3} \cdot \frac{1}{x^5 (e^x - 1)}
\]

Now we need to find the maximum of this function with respect to \( x \). 

### 4. **Differentiate and Solve:**

Take the derivative of \( I(\lambda, T) \) with respect to \( x \):

\[
\frac{d}{dx} \left( \frac{1}{x^5 (e^x - 1)} \right) = 0
\]

After differentiating, you will get a transcendental equation of the form:

\[
5(1 - e^{-x}) = x e^{-x}
\]

Solving this equation numerically gives:

\[
x_{\text{max}} \approx 4.965
\]

### 5. **Substitute Back for \( \lambda_{\text{max}} \):**

Recalling that \( x = \frac{hc}{\lambda k_B T} \), we can solve for \( \lambda_{\text{max

Explore the detailed derivation of Wien's Displacement Law from Planck's Law, including the use of calculus to find the peak wavelength of blackbody radiation. This tutorial walks through the step-by-step process, featuring key formulas, such as Planck's Law and Wien's Displacement Law. Suitable for advanced undergraduate and graduate physics students.

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