Derive Wien's displacement law.

Steps to Solve

1. Start with Planck's Law

We begin with Planck's law for black-body radiation, which states that the spectral radiance of a black body at temperature $T$ is given by:

$$ B(\lambda, T) = \frac{2hc^2}{\lambda^5} \cdot \frac{1}{e^{\frac{hc}{\lambda kT}} - 1} $$

where $B(\lambda, T)$ is the spectral radiance, $h$ is Planck's constant, $c$ is the speed of light, $\lambda$ is the wavelength, and $k$ is the Boltzmann constant.

2. Find the maximum wavelength

To derive Wien's displacement law, we need to find the wavelength $\lambda_{max}$ at which the radiation is maximized. This involves differentiating Planck’s law with respect to $\lambda$ and setting the derivative to zero:

$$ \frac{dB(\lambda, T)}{d\lambda} = 0 $$

3. Apply implicit differentiation

Using implicit differentiation, we simplify the expression and solve for $\lambda$. This might involve using the quotient rule and simplifying the terms in the derivative until we find:

$$ \frac{5}{\lambda} - \frac{hc}{\lambda^2 kT} e^{\frac{hc}{\lambda kT}} = 0 $$

4. Rearrange the equation

By rearranging the equation, we eventually derive that:

$$ 5 = \frac{hc}{\lambda kT} e^{\frac{hc}{\lambda kT}} $$

5. Logarithmic properties

Taking the natural log of both sides to aid in solving for $\lambda$ and simplifying the resulting equation leads us toward establishing the relationship between temperature and wavelength.

6. Final expression for Wien's law

After simplifying and rearranging, we ultimately arrive at Wien's displacement law expressed as:

$$ \lambda_{max} = \frac{b}{T} $$

where $b$ is a constant known as Wien's displacement constant, approximately equal to $2898 , \mu m \cdot K$.