To determine the wavelength of yellow light with a frequency of [tex]\(5.2 \times 10^{14}\)[/tex] Hz, follow these steps:
1. Understand the relationship:
The relationship between the speed of light ([tex]\(c\)[/tex]), frequency ([tex]\(f\)[/tex]), and wavelength ([tex]\(\lambda\)[/tex]) is given by the equation:
[tex]\[
\lambda = \frac{c}{f}
\][/tex]
where:
- [tex]\(\lambda\)[/tex] is the wavelength,
- [tex]\(c\)[/tex] is the speed of light, and
- [tex]\(f\)[/tex] is the frequency.
2. Given values:
- Frequency ([tex]\(f\)[/tex]) = [tex]\(5.2 \times 10^{14}\)[/tex] Hz
- Speed of light ([tex]\(c\)[/tex]) = [tex]\(3.0 \times 10^8\)[/tex] m/s
3. Plug in the given values:
[tex]\[
\lambda = \frac{3.0 \times 10^8 \text{ m/s}}{5.2 \times 10^{14} \text{ Hz}}
\][/tex]
4. Calculate the division step-by-step:
- Numerator: [tex]\(3.0 \times 10^8\)[/tex] m/s
- Denominator: [tex]\(5.2 \times 10^{14}\)[/tex] Hz
5. Perform the division:
First, divide the coefficients:
[tex]\[
\frac{3.0}{5.2} \approx 0.576923
\][/tex]
Then, subtract the exponents of the powers of 10:
[tex]\[
10^8 \div 10^{14} = 10^{8-14} = 10^{-6}
\][/tex]
6. Combine the results:
[tex]\[
\lambda = 0.576923 \times 10^{-6} \text{ m}
\][/tex]
7. Express in scientific notation:
Rewrite 0.576923 as [tex]\(5.769230769230769\)[/tex] and adjust the exponent:
[tex]\[
\lambda = 5.769230769230769 \times 10^{-7} \text{ m}
\][/tex]
So, the wavelength of yellow light with a frequency of [tex]\(5.2 \times 10^{14}\)[/tex] Hz is approximately [tex]\(5.769230769230769 \times 10^{-7}\)[/tex] m.
