To determine the wavelength of a photon with a given frequency, we can use the relationship between the speed of light, frequency, and wavelength. This relationship is given by the formula:
[tex]\[ c = \lambda \cdot f \][/tex]
where:
- [tex]\( c \)[/tex] is the speed of light in a vacuum, approximately [tex]\( 3 \times 10^8 \)[/tex] meters per second.
- [tex]\( \lambda \)[/tex] (lambda) is the wavelength in meters.
- [tex]\( f \)[/tex] is the frequency in hertz (Hz).
We can rearrange this formula to solve for the wavelength ([tex]\( \lambda \)[/tex]):
[tex]\[ \lambda = \frac{c}{f} \][/tex]
Given:
- The frequency [tex]\( f = 4.72 \times 10^{14} \)[/tex] Hz.
- The speed of light [tex]\( c = 3 \times 10^8 \)[/tex] meters per second.
Let's substitute the values into the formula to find the wavelength in meters:
[tex]\[ \lambda = \frac{3 \times 10^8 \, \text{m/s}}{4.72 \times 10^{14} \, \text{Hz}} \][/tex]
This yields:
[tex]\[ \lambda \approx 6.36 \times 10^{-7} \, \text{meters} \][/tex]
Next, we need to convert the wavelength from meters to nanometers. Since there are [tex]\( 1 \times 10^9 \)[/tex] nanometers in a meter, we multiply the wavelength by [tex]\( 10^9 \)[/tex] to convert it to nanometers:
[tex]\[ \lambda \approx 6.36 \times 10^{-7} \, \text{meters} \times 10^9 \, \frac{\text{nanometers}}{\text{meter}} \][/tex]
[tex]\[ \lambda \approx 636 \, \text{nanometers} \][/tex]
Therefore, the wavelength of the photon with a frequency of [tex]\( 4.72 \times 10^{14} \)[/tex] Hz is closest to:
D. 635 nm
