To find the wavelength of a yellow light with a given frequency, we use the fundamental relationship between the speed of light, frequency, and wavelength. The formula for this relationship is:
[tex]\[ \lambda = \frac{c}{f} \][/tex]
where:
- [tex]\(\lambda\)[/tex] is the wavelength
- [tex]\(c\)[/tex] is the speed of light
- [tex]\(f\)[/tex] is the frequency
Given:
- The frequency [tex]\(f\)[/tex] of the yellow light is [tex]\(5.2 \times 10^{14}\)[/tex] Hz
- The speed of light [tex]\(c\)[/tex] is [tex]\(3.0 \times 10^8\)[/tex] m/s
Step-by-step procedure to calculate the wavelength:
1. Set up the formula with the given values:
[tex]\[
\lambda = \frac{3.0 \times 10^8 \, \text{m/s}}{5.2 \times 10^{14} \, \text{Hz}}
\][/tex]
2. Divide the coefficients:
[tex]\[
\lambda = \frac{3.0}{5.2} \times \frac{10^8}{10^{14}}
\][/tex]
3. Simplify the coefficient division:
[tex]\[
\frac{3.0}{5.2} \approx 0.576923076923
\][/tex]
4. Simplify the exponent division:
[tex]\[
\frac{10^8}{10^{14}} = 10^{8-14} = 10^{-6}
\][/tex]
5. Combine the simplified coefficient and exponent:
[tex]\[
\lambda \approx 0.576923076923 \times 10^{-6} \, \text{meters}
\][/tex]
6. Express the result in scientific notation:
[tex]\[
0.576923076923 \approx 5.769230769230769 \times 10^{-1}
\][/tex]
Then:
[tex]\[
0.576923076923 \times 10^{-6} = 5.769230769230769 \times 10^{-1} \times 10^{-6} = 5.769230769230769 \times 10^{-7} \, \text{meters}
\][/tex]
Therefore, the wavelength of the yellow light is:
[tex]\[ \boxed{5.769230769230769 \times 10^{-7} \, \text{meters} } \][/tex]
Thus, the wavelength is approximately [tex]\(5.769230769230769 \times 10^{-7}\)[/tex] meters.
