To find the wavelength of an infrared wave traveling through a vacuum, given its frequency, we can use the basic relationship between the speed of light ([tex]\(c\)[/tex]), frequency ([tex]\(f\)[/tex]), and wavelength ([tex]\(\lambda\)[/tex]). The formula that relates these quantities is:
[tex]\[ c = \lambda \cdot f \][/tex]
Where:
- [tex]\( c \)[/tex] is the speed of light in a vacuum ([tex]\( 3.0 \times 10^8 \)[/tex] meters per second).
- [tex]\( f \)[/tex] is the frequency of the wave ([tex]\( 4.0 \times 10^{14} \)[/tex] Hertz).
- [tex]\( \lambda \)[/tex] is the wavelength, which we need to find.
To isolate the wavelength ([tex]\(\lambda\)[/tex]), we can rearrange the formula:
[tex]\[ \lambda = \frac{c}{f} \][/tex]
Now we can substitute the known values into the formula:
[tex]\[ \lambda = \frac{3.0 \times 10^8 \, \text{m/s}}{4.0 \times 10^{14} \, \text{Hz}} \][/tex]
Performing the division:
[tex]\[ \lambda = \frac{3.0}{4.0} \times 10^8 \times 10^{-14} \, \text{m} \][/tex]
[tex]\[ \lambda = 0.75 \times 10^{-6} \, \text{m} \][/tex]
Expressing [tex]\(0.75\)[/tex] as a fraction of [tex]\(1\)[/tex] in scientific notation:
[tex]\[ \lambda = 7.5 \times 10^{-7} \, \text{m} \][/tex]
Hence, the wavelength of the infrared wave is [tex]\(7.5 \times 10^{-7}\)[/tex] meters. Therefore, the correct answer is:
[tex]\[ 7.5 \times 10^{-7} \, \text{m} \][/tex]
