To determine the wavelength of a photon with a given frequency, we can use the relationship between the speed of light, frequency, and wavelength. The formula is:
[tex]\[ c = \lambda \times f \][/tex]
where:
- [tex]\( c \)[/tex] is the speed of light, which is [tex]\( 3 \times 10^8 \)[/tex] meters per second (m/s),
- [tex]\( \lambda \)[/tex] is the wavelength in meters,
- [tex]\( f \)[/tex] is the frequency in hertz (Hz).
Given:
- The frequency [tex]\( f = 4.72 \times 10^{14} \)[/tex] Hz,
- The speed of light [tex]\(c = 3 \times 10^8 \)[/tex] m/s.
Let's solve for the wavelength ([tex]\( \lambda \)[/tex]):
1. Rearrange the formula to solve for [tex]\( \lambda \)[/tex]:
[tex]\[ \lambda = \frac{c}{f} \][/tex]
2. Substitute the known values into the formula:
[tex]\[ \lambda = \frac{3 \times 10^8 \, \text{m/s}}{4.72 \times 10^{14} \, \text{Hz}} \][/tex]
3. Perform the division:
[tex]\[ \lambda \approx 6.355932203389831 \times 10^{-7} \, \text{m} \][/tex]
4. Convert the wavelength from meters to nanometers (nm), noting that [tex]\(1 \text{ meter} = 10^9 \text{ nanometers}\)[/tex]:
[tex]\[ \lambda_{\text{nm}} = 6.355932203389831 \times 10^{-7} \, \text{m} \times 10^9 \, \text{nm/m} \][/tex]
[tex]\[ \lambda_{\text{nm}} \approx 635.5932203389831 \, \text{nm} \][/tex]
Now that we have the wavelength in nanometers, we can compare it to the given options:
- A. 421 nm
- B. 142 nm
- C. 635 nm
- D. 313 nm
The calculated wavelength is approximately [tex]\( 635.59 \)[/tex] nm, which is closest to option C.
Therefore, the wavelength of the photon with a frequency of [tex]\( 4.72 \times 10^{14} \)[/tex] Hz is [tex]\( \boxed{635 \text{ nm}} \)[/tex].
