Certainly! Let's solve this step by step using the given information.
We have the relationship between frequency, speed of light, and wavelength given by the formula:
[tex]\[ \lambda = \frac{v}{f} \][/tex]
where:
- [tex]\(\lambda\)[/tex] is the wavelength,
- [tex]\(v\)[/tex] is the speed of light, and
- [tex]\(f\)[/tex] is the frequency.
Given:
[tex]\[ v = 3 \times 10^8 \, \text{m/s} \][/tex]
[tex]\[ f = 8 \times 10^{14} \, \text{Hz} \][/tex]
We substitute the given values into the formula:
[tex]\[ \lambda = \frac{3 \times 10^8 \, \text{m/s}}{8 \times 10^{14} \, \text{Hz}} \][/tex]
Carrying out the division:
[tex]\[ \lambda = \frac{3 \times 10^8}{8 \times 10^{14}} \][/tex]
This simplifies to:
[tex]\[ \lambda = \frac{3}{8} \times 10^{8-14} \][/tex]
[tex]\[ \lambda = \frac{3}{8} \times 10^{-6} \][/tex]
[tex]\[ \lambda = 0.375 \times 10^{-6} \][/tex]
Expressing it in scientific notation:
[tex]\[ \lambda = 3.75 \times 10^{-7} \, \text{m} \][/tex]
So the wavelength of the electromagnetic wave is:
[tex]\[ \lambda = 3.75 \times 10^{-7} \, \text{m} \][/tex]
Comparing this with the given options:
A. [tex]\(3.8 \times 10^{-7} \, \text{m}\)[/tex]
B. [tex]\(2.7 \times 10^{-6} \, \text{m}\)[/tex]
C. [tex]\(2.7 \times 10^6 \, \text{m}\)[/tex]
D. [tex]\(3.8 \times 10^{21} \, \text{m}\)[/tex]
The correct answer is closest to option A. [tex]\(3.8 \times 10^{-7} \, \text{m}\)[/tex].
Therefore, the answer is:
A. [tex]\(3.8 \times 10^{-7} \, \text{m}\)[/tex].
