To find the wavelength of light given its frequency, we can use the fundamental relationship between the speed of light ([tex]\( c \)[/tex]), the frequency ([tex]\( f \)[/tex]), and the wavelength ([tex]\( \lambda \)[/tex]):
[tex]\[ \lambda = \frac{c}{f} \][/tex]
Where:
- [tex]\( \lambda \)[/tex] is the wavelength,
- [tex]\( c \)[/tex] is the speed of light in a vacuum, approximately [tex]\( 3 \times 10^8 \)[/tex] meters per second,
- [tex]\( f \)[/tex] is the frequency of the light.
Given:
- The frequency [tex]\( f = 7.21 \times 10^{14} \)[/tex] Hz,
- The speed of light [tex]\( c = 3 \times 10^8 \)[/tex] m/s.
We can plug in these values to calculate the wavelength in meters:
[tex]\[ \lambda = \frac{3 \times 10^8 \text{ m/s}}{7.21 \times 10^{14} \text{ Hz}} \][/tex]
Calculating this division:
[tex]\[ \lambda = 4.160887656033287 \times 10^{-7} \text{ meters} \][/tex]
Next, we need to convert the wavelength from meters to nanometers. Since 1 meter is equal to [tex]\( 10^9 \)[/tex] nanometers:
[tex]\[ \lambda_{\text{nm}} = 4.160887656033287 \times 10^{-7} \text{ meters} \times 10^9 \text{ nm/m} \][/tex]
[tex]\[ \lambda_{\text{nm}} = 416.0887656033287 \text{ nm} \][/tex]
Rounding to the nearest whole number:
[tex]\[ \lambda \approx 416 \text{ nm} \][/tex]
So the wavelength of the light with a frequency of [tex]\( 7.21 \times 10^{14} \)[/tex] Hz is approximately 416 nm. Therefore, the correct answer is:
B. 416 nm
