Chapter 39, Problem 1P

(a)

To determine

The order of magnitude of the wavelength of the thermally produced photons in lightning and thunder.

Answer to Problem 1P

The order of magnitude of the wavelength of the thermally produced photons in lightning is $\approx {10}^{-7}\text{m}$ and for thunder is $\approx {10}^{-10}\text{m}$.

Explanation of Solution

Write the expression for Wein’s displacement law.

    ${\lambda }_{\max }T=2.898\times {10}^{-3}\text{m}\cdot \text{K}$                                                                                   (I)

Here, ${\lambda }_{\max }$ is the maximum wavelength and $T$ is the temperature.

Rewrite equation (I) to find the order of the wavelength.

    ${\lambda }_{\max }=\frac{2.898\times {10}^{-3}\text{m}\cdot \text{K}}{T}$                                                                                  (II)

Conclusion:

Substitute ${10}^4\text{K}$ for $T$ in equation (II) to find the order of the wavelength in case of lightning.

    $\begin{array}{c}{\lambda }_{\max }=\frac{2.898\times {10}^{-3}\text{m}\cdot \text{K}}{{10}^4\text{K}}\\ =2.898\times {10}^{-7}\text{m}\\ \approx {\text{10}}^{-7}\text{m}\end{array}$

Substitute ${10}^7\text{K}$ for $T$ in equation (II) to find the order of the wavelength in case of thunder.

    $\begin{array}{c}{\lambda }_{\max }=\frac{2.898\times {10}^{-3}\text{m}\cdot \text{K}}{{10}^7\text{K}}\\ =2.898\times {10}^{-10}\text{m}\\ \approx {\text{10}}^{-10}\text{m}\end{array}$

Thus, the order of magnitude of the wavelength of the thermally produced photons in lightning is $\approx {10}^{-7}\text{m}$ and for thunder is $\approx {10}^{-10}\text{m}$.

(b)

To determine

The region in the spectrum where the lightning and thunder will radiate more strongly.

Answer to Problem 1P

The lightning radiate more strongly in the ultraviolet region whereas the thunder radiate more strongly in the x-ray and the gamma ray region.

Explanation of Solution

Lightning have the wavelength in the order $\approx {10}^{-7}\text{m}$. This falls in the ultraviolet region of the spectrum.

Thunder have the wavelength in the order $\approx {10}^{-10}\text{m}$. This falls in the x-rays and gamma rays region of the spectrum.

Conclusion:

Thus, the lightning radiate more strongly in the ultraviolet region whereas the thunder radiate more strongly in the x-ray and the gamma ray region.