To find the wavelength of a photon with a given frequency, we use the relationship between the speed of light, frequency, and wavelength. This relationship is given by the formula:
[tex]\[ \text{wavelength} (\lambda) = \frac{\text{speed of light} (c)}{\text{frequency} (f)} \][/tex]
Where:
- The speed of light ([tex]\(c\)[/tex]) is [tex]\(3 \times 10^8 \, \text{m/s}\)[/tex].
- The frequency ([tex]\(f\)[/tex]) is [tex]\(6.56 \times 10^{14} \, \text{Hz}\)[/tex].
First, we calculate the wavelength in meters:
[tex]\[ \lambda = \frac{3 \times 10^8 \, \text{m/s}}{6.56 \times 10^{14} \, \text{Hz}} \][/tex]
The resulting wavelength is:
[tex]\[ \lambda = 4.573170731707317 \times 10^{-7} \, \text{m} \][/tex]
Since this value is in meters, we need to convert it to nanometers (nm). We know that:
[tex]\[ 1 \, \text{meter} = 10^9 \, \text{nanometers} \][/tex]
So, we multiply the wavelength in meters by [tex]\(10^9\)[/tex]:
[tex]\[ \lambda_{\text{nm}} = 4.573170731707317 \times 10^{-7} \, \text{m} \times 10^9 \, \text{nm/m} \][/tex]
This gives us the wavelength in nanometers:
[tex]\[ \lambda_{\text{nm}} = 457.3170731707317 \, \text{nm} \][/tex]
Therefore, the wavelength of a photon with a frequency of [tex]\(6.56 \times 10^{14} \, \text{Hz}\)[/tex] is approximately 457 nm.
So the correct answer is:
C. 457 nm
