Chapter 30, Problem 1CQ

To determine

The brief description of the Ultraviolet Catastrophe.

Answer to Problem 1CQ

Ultraviolet Catastrophe is an event causing great and usually sudden damage or suffering when an ideal black body at thermal equilibrium emits radiation with infinite power.

Explanation of Solution

Ultraviolet Catastrophe is a classical prediction and is also known as Rayleigh–Jeans catastrophe.

Here the term Ultraviolet refers to the fact that the problem appears in short wavelength region of electromagnetic spectrum whereas the term Catastrophe refers to n event causing great and usually sudden damage or suffering.

Ultraviolet Catastrophe predicts that an ideal black body at thermal equilibrium will emit radiation in all frequency ranges and it emits more energy when the value of frequency increases with infinite power.

The expression for the energy is,

$E_{\text{λ}}=\frac{8\pi kT}{{\lambda }^4}$

Here,

$k$ is the Boltzmann Constant.

$T$ is the temperature.

$\lambda$ is the wavelength of electromagnetic wave.

The above equation can explain the energy distribution curves for longer wavelengths satisfactorily, but fails to explain the curves for smaller wavelengths.

Integrate the above equation from zero to infinity.

$\begin{array}{c}{\displaystyle \underset{0}{\overset{\infty }{\int }}{E}_{\text{λ}}}={\displaystyle \underset{0}{\overset{\infty }{\int }}\frac{8\pi kT}{{\lambda }^4}}\\ =\infty \end{array}$

Conclusion:

Therefore, Ultraviolet Catastrophe is an event causing great and usually sudden damage or suffering when an ideal black body at thermal equilibrium emits radiation with infinite power.