Derive Wien's displacement law.
Understand the Problem
The question is asking to derive Wien's displacement law, which relates to the temperature of a black body and the wavelength at which its emission is maximized. This involves concepts from thermodynamics and quantum mechanics.
Answer
Wien's displacement law is given by $ \lambda_{max} = \frac{b}{T} $, with $ b \approx 2898 \, \mu m \cdot K $.
Answer for screen readers
Wien's displacement law is expressed as:
$$ \lambda_{max} = \frac{b}{T} $$
where $b \approx 2898 , \mu m \cdot K$.
Steps to Solve
- 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.
- 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 Planckâ€™s law with respect to $\lambda$ and setting the derivative to zero:
$$ \frac{dB(\lambda, T)}{d\lambda} = 0 $$
- 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:
$$ \frac{5}{\lambda} - \frac{hc}{\lambda^2 kT} e^{\frac{hc}{\lambda kT}} = 0 $$
- Rearrange the equation
By rearranging the equation, we eventually derive that:
$$ 5 = \frac{hc}{\lambda kT} e^{\frac{hc}{\lambda kT}} $$
- 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.
- 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 $2898 , \mu m \cdot K$.
Wien's displacement law is expressed as:
$$ \lambda_{max} = \frac{b}{T} $$
where $b \approx 2898 , \mu m \cdot K$.
More Information
Wien's displacement law explains how the peak wavelength of radiation emitted by a black body decreases as its temperature increases, highlighting the relationship between thermal radiation and temperature.
Tips
- Ignoring Constants: A common mistake is neglecting constants such as $h$, $c$, and $k$ when differentiating. Accurate use of these constants is crucial for obtaining the correct relation.
- Incorrectly Handling Exponentials: Misapplication of the properties of exponential functions can lead to errors in simplification. Always double-check the steps involving exponentials during differentiation.