Chapter 5.3, Problem 16SP

Interpretation Introduction

Interpretation: The frequency of radiation of a given wavelength needs to be determined. Also, the region in which this radiation should be in the electromagnetic spectrum needs to be identified.

Concept Introduction: The relation between wavelength and frequency of radiation is as follows:

  $v=\frac{c}{\lambda }$

Here, $v$ is the frequency of radiation, $\lambda$ is the wavelength of radiation, and c is the speed of light. The value of the speed of light is constant that is $3\times {10}^8\text{ m/s}$ .

Answer to Problem 16SP

  $6.0\times {10}^{15}\text{ Hz}$ , UV region.

Explanation of Solution

From the given wavelength of radiation, the frequency can be calculated as follows:

  $v=\frac{c}{\lambda }$

Here, the value of the speed of light is constant that is $3\times {10}^8\text{ m/s}$ and the wavelength of radiation is given $5.0\times {10}^{-8}\text{ m}$ .

Substitute the values in the above formula,

  $\begin{array}{c}v=\frac{3\times {10}^8\text{ m/s}}{5\times {10}^{-8}\text{ m}}\\ =6.0\times {10}^{15}{\text{ s}}^{-1}\end{array}$

Here,

  $1\text{ Hz}=1{\text{ s}}^{-1}$

Thus,

  $\begin{array}{c}v=6.0\times {10}^{15}{\text{ s}}^{-1}\left(\frac{1\text{ Hz}}{1{\text{ s}}^{-1}}\right)\\ =6.0\times {10}^{15}\text{ Hz}\end{array}$

The below table shows the range of frequency for all the regions in the electromagnetic spectrum:

Region	Frequency range in Hz
Gamma-rays	${10}^{21}-{10}^{19}$
X-rays	${10}^{19}-{10}^{17}$
Ultraviolet (UV)	${10}^{17}-{10}^{15}$
Visible light	${10}^{15}$
Infra-red	${10}^{14}-{10}^{12}$
Microwave	${10}^{12}-{10}^9$
Radio wave	${10}^9-{10}^7$

From the above table, it can be seen that the calculated frequency lies in the range of ${10}^{17}-{10}^{15}$ ; thus, it comes under the UV region.

Conclusion

The frequency of radiation of a given wavelength is $6.0\times {10}^{15}\text{ Hz}$ . The radiation should be in the UV region in the electromagnetic spectrum.