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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

  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.

  1. 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:

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

  1. 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:

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

  1. Rearrange the equation

By rearranging the equation, we eventually derive that:

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

  1. 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.

  1. 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.
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