Planck's Law describes the spectral radiance of electromagnetic radiation emitted by a black body in thermal equilibrium at a given temperature \( T \). The law is expressed as:

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

Where:
- \( B(\lambda, T) \) is the spectral radiance (power per unit area per unit wavelength per unit solid angle),
- \( h \) is Planck's constant,
- \( c \) is the speed of light in a vacuum,
- \( \lambda \) is the wavelength,
- \( k_B \) is Boltzmann's constant,
- \( T \) is the absolute temperature of the black body.

### Step-by-Step Derivation of Planck's Law

To find the derivative of Planck's Law with respect to \( \lambda \), we will differentiate the equation above using the product rule, quotient rule, and chain rule.

We have:

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

### Derivative \( \frac{dB(\lambda, T)}{d\lambda} \)

First, let’s break the function into two parts:
1. \( A(\lambda) = \frac{2hc^2}{\lambda^5} \),
2. \( f(\lambda) = \frac{1}{e^{\frac{hc}{\lambda k_B T}} - 1} \).

Using the product rule:
\[
\frac{dB(\lambda, T)}{d\lambda} = \frac{dA(\lambda)}{d\lambda} f(\lambda) + A(\lambda) \frac{df(\lambda)}{d\lambda}
\]

#### 1. Derivative of \( A(\lambda) = \frac{2hc^2}{\lambda^5} \):

\[
\frac{dA(\lambda)}{d\lambda} = -5 \cdot \frac{2hc^2}{\lambda^6} = -\frac{10hc^2}{\lambda^6}
\]

#### 2. Derivative of \( f(\lambda) = \frac{1}{e^{\frac{hc}{\lambda k_B T}} - 1} \):

Now, apply the chain rule to \( f(\lambda) \):
\[
\frac{df(\lambda)}{d\lambda} = -\frac{e^{\frac{hc}{\lambda k_B T}} \cdot \frac{hc}{\lambda^2 k_B T}}{\left(e^{\frac{hc}{\lambda k_B T}} - 1\right)^2}
\]

#### Full Derivative of \( B(\lambda, T) \):

Putting it all together:
\[
\frac{dB(\lambda, T)}{d\lambda} = -\frac{10hc^2}{\lambda^6} \cdot \frac{1}{e^{\frac{hc}{\lambda k_B T}} - 1} + \frac{2hc^2}{\lambda^5} \cdot \left( -\frac{e^{\frac{hc}{\lambda k_B T}} \cdot \frac{hc}{\lambda^2 k_B T}}{\left(e^{\frac{hc}{\lambda k_B T}} - 1\right)^2} \right)
\]

This is the derivative of Planck's Law with respect to \( \lambda \), where each term accounts for the changes in radiance with the wavelength.

Would you like a more detailed explanation of any specific part?

Here are some related questions:
1. What physical meaning does the derivative of Planck's law have?
2. How does the Wien's displacement law relate to the derivative of Planck's law?
3. What is the significance of Planck's constant in black-body radiation?
4. Can you explain the Stefan-Boltzmann law and how it derives from Planck's law?
5. How does Planck's law behave at short and long wavelengths?

**Tip:** The peak wavelength of black body radiation can be found by setting the derivative of Planck’s law to zero and solving for \( \lambda \), which leads to Wien's displacement law.

Discover the detailed derivation of Planck's law and learn how to differentiate the spectral radiance equation with respect to wavelength. This guide breaks down the steps using the product rule and chain rule, and explains the meaning of key constants such as Planck's constant and Boltzmann's constant.

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