Question

BIO Medical X rays. Medical x rays are taken with electromagnetic waves having a wavelength of around 0.10 nm in air. What are the frequency, period, and wave number of such waves?

Short answer

The frequency, time period, and wave number of waves are$3\times {10}^8\mathrm{Hz},3.3\times {10}^{-19})\sec$ and $6.3\times {10}^{10}\mathrm{rad}/\mathrm{m}$.

Step-by-step solution

Define frequency and write formulas.

The number of waves that moves a fixed point in unit time is known as frequency.

$f=\frac{1}{T}$

Where,T =Time period in s.

The frequency, wavelength, and speed of light of any wave are related by the wavelength-frequency relationship.

The wavelength-frequency relationship is:

$c=\lambda f$

Where, c is the speed of light which is equal to $3\times {10}^8\mathrm{m}/\mathrm{s}$,$\lambda$ is the wavelength andf is the frequency.

The wavelength is inversely proportional to the frequency.

$\lambda =\frac{c}{f}$or $\frac{v}{f}$

Where, v is velocity of wave in m/s.

The relation between wave number and wavelength is:

$k=\frac{2\mathrm{\pi }}{\lambda }$

Where, k is wavenumber in rad/m.

Determine the frequency of wave.

Given that,

$\mathrm{\lambda }=0.10\times {10}^{-9}\mathrm{m}$

The wavelength-frequency relationship is:

$$\begin{gathered} c=\lambda f \\ \Rightarrow f=\frac{c}{\lambda } \end{gathered}$$

Substitute the values of variables

$$\begin{gathered} \mathrm{f}=\frac{\mathrm{c}}{\mathrm{\lambda }} \\ =\frac{3\times {10}^8}{0.10\times {10}^9} \\ =3\times {10}^{18}\mathrm{Hz} \end{gathered}$$

Hence, the frequency of waves is $3\times {10}^{18}\mathrm{Hz}$.

Determine the time period.

The frequency of wave is:

$$\begin{gathered} f=\frac{1}{T} \\ \Rightarrow T=\frac{1}{f} \end{gathered}$$

Substitute the values of variables

$$\begin{gathered} T=\frac{1}{f} \\ =\frac{1}{3\times {10}^{19}} \\ =3.3\times {10}^{-19}sec \end{gathered}$$

Hence the time period is equal to $3.3\times {10}^{-19}\sec$.

Determine the wave number.

The relation between wave number and wavelength is:

$k=\frac{2\mathrm{\pi }}{\lambda }$

Substitute the values of variables

$$\begin{gathered} \mathrm{k}=\frac{2\mathrm{\pi }}{0.10\times {10}^{-9}} \\ =6.3\times {10}^{10}\mathrm{rad}/\mathrm{m} \end{gathered}$$

Hence, the wave number is equal to $6.3\times {10}^{10}\mathrm{rad}/\mathrm{m}$.