To solve the problem of finding the wavelength of an electromagnetic (EM) wave with a given frequency of [tex]\(10^{17} \, \text{Hz}\)[/tex], we follow these step-by-step instructions:
1. Identify the given values:
- Speed of light, [tex]\( c \)[/tex] [tex]\( \approx 3 \times 10^8 \, \text{m/s} \)[/tex]
- Frequency of the wave, [tex]\( f = 10^{17} \, \text{Hz} \)[/tex]
2. Use the relationship between speed, frequency, and wavelength:
The wavelength ([tex]\( \lambda \)[/tex]) can be calculated using the formula:
[tex]\[
\lambda = \frac{c}{f}
\][/tex]
3. Plug in the values to calculate the wavelength in meters:
[tex]\[
\lambda = \frac{3 \times 10^8 \, \text{m/s}}{10^{17} \, \text{Hz}}
\][/tex]
[tex]\[
\lambda = \frac{3 \times 10^8}{10^{17}}
\][/tex]
[tex]\[
\lambda = 3 \times 10^{-9} \, \text{m}
\][/tex]
4. Convert the wavelength from meters to nanometers:
Because [tex]\(1 \, \text{nm} = 10^{-9} \, \text{m}\)[/tex],
[tex]\[
\lambda = 3 \times 10^{-9} \, \text{m} = 3 \, \text{nm}
\][/tex]
So, the wavelength of the EM wave with a frequency of [tex]\(10^{17} \, \text{Hz}\)[/tex] is 3 nanometers (3 nm). Therefore, the correct option is:
- 3 nm
