Question

The speed of electromagnetic waves (which include visible light, radio, and x rays) in vacuum is$\mathbf{30}\mathit{x}{\mathbf{10}}^{\mathbf{8}}\mathbf{}\mathit{m}\mathbf{/}\mathit{s}$. (a) Wavelengths of visible light waves range from about 700 nmin the violet to about 700 nmin the red. What is the range of frequencies of these waves? (b)The range of frequencies for shortwave radio (for example, FM radio and VHF television) is 1.5to 300 MHz . What is the corresponding wavelength range? (c) X-ray wavelengths range from about 5.0nmto about$\mathbf{1}\mathbf{.}\mathbf{0}\mathit{x}{\mathbf{10}}^{\mathbf{-}\mathbf{2}}\mathbf{}\mathit{n}\mathit{m}$ . What is the frequency range for x rays?

Short answer

1. Range of frequencies is $4.3\times {10}^{14}\mathrm{Hz}\mathrm{and}7.5\times {10}^{14}\mathrm{Hz}$
2. Wavelength range is$1.0\mathrm{m}\mathrm{and}2.0\times {10}^2\mathrm{m}$
3. Frequency range for x rays is $6.0\times {10}^{16}\mathrm{Hz}\mathrm{and}3.0\times {10}^{19}\mathrm{Hz}$

Step-by-step solution

Given data

Speed of electromagnetic wave,$c=3.00\times {10}^{8}m/s$

Wavelength of visible${\lambda }_1=400nmor400\times {10}^{-9}m$

Wavelength of visible light${\lambda }_2=700nmor700\times {10}^{-9}m$

Frequency ranges,$f=1.5to300MHz$

X ray wavelength range is $5.0nmor1.0\times {10}^{-2}nm$

Understanding the concept of resonant frequency

Here we have to use the basic formula for wave velocity in terms of wavelength and frequency.

Formula:

The speed of light, $c=f\lambda ....\dots \left(1\right)$

Step 3(a): Calculation of range of frequencies

Using equation (1), we get the value of maximum frequency of visible light for first wavelength as:

$$\begin{gathered} f_1=\frac{v}{{\lambda }_1} \\ =\frac{3.00\times {10}^8}{400\times {10}^{-9}} \\ =7.5\times {10}^{14}Hz \end{gathered}$$

Now minimum frequency$f_2$using equation (1) is given as follows:

localid="1660992239723" $$\begin{gathered} f_2=\frac{v}{{\lambda }_2} \\ =\frac{3.00\times {10}^8}{700\times {10}^{-9}} \\ =4.3\times {10}^{14}Hz \end{gathered}$$.

Hence, the range of frequencies is $4.3\times {10}^{14}Hzand7.5\times {10}^{14}Hz$

Step 4(b): Calculation of wavelength range

Now using minimum frequency and equation (1), we get the wavelength${\lambda }_1$as:

$\begin{array}{l}{\lambda }_1=\frac{3.00\times {10}^8}{300\times {10}^6}\\ =1.0m\end{array}$.

Now using maximum frequency and equation (1), we get the wavelength${\lambda }_2$as:

$\begin{array}{l}{\lambda }_2=\frac{3.00\times {10}^8}{1.5\times {10}^6}\\ =200m\end{array}$

Hence, the value of wavelength range is $1.0\mathrm{m}\mathrm{and}2.0\times {10}^2\mathrm{m}$

Step 5(c): Calculation of frequency range for x-rays

Now minimum frequency$f_1$using equation (1) for x-rays is given as:

$\begin{array}{l}{f}_1=\frac{3.00\times {10}^8}{5\times {10}^{-9}}\\ =6.0\times {10}^{16}Hz\end{array}$

Now maximum frequency$f_2$using equation (1) for x-rays is given as:

$\begin{array}{l}{f}_2=\frac{3.00\times {10}^8}{1.0\times {10}^{-11}}\\ =3.0\times {10}^{19}Hz\end{array}$

Hence, the value of frequency range for x rays is$6.0\times {10}^{16}Hzand3.0\times {10}^{19}Hz$