To find the wavelength of an infrared wave traveling through a vacuum given its frequency, we can use the equation relating the speed of light, frequency, and wavelength. The speed of light [tex]\( c \)[/tex], frequency [tex]\( f \)[/tex], and wavelength [tex]\( \lambda \)[/tex] are related by the formula:
[tex]\[ \lambda = \frac{c}{f} \][/tex]
where:
- [tex]\( c \)[/tex] is the speed of light, approximately [tex]\( 3.0 \times 10^8 \, \text{m/s} \)[/tex]
- [tex]\( f \)[/tex] is the frequency of the wave
Given the frequency [tex]\( f = 4.0 \times 10^{14} \, \text{Hz} \)[/tex], we can plug these values into the formula to find the wavelength [tex]\( \lambda \)[/tex].
[tex]\[ \lambda = \frac{3.0 \times 10^8 \, \text{m/s}}{4.0 \times 10^{14} \, \text{Hz}} \][/tex]
Carrying out the division:
[tex]\[ \lambda = \frac{3.0}{4.0} \times 10^{8 - 14} \][/tex]
[tex]\[ \lambda = 0.75 \times 10^{-6} \, \text{m} \][/tex]
This can be rewritten in scientific notation:
[tex]\[ \lambda = 7.5 \times 10^{-7} \, \text{m} \][/tex]
Thus, the wavelength of the infrared wave is [tex]\( 7.5 \times 10^{-7} \, \text{m} \)[/tex].
From the given choices, the correct answer is:
[tex]\[ 7.5 \times 10^{-7} \, \text{m} \][/tex]
