Answer :
To determine the wavelength of yellow light with a frequency of [tex]\(5.2 \times 10^{14} \, \text{Hz}\)[/tex] given the speed of light in a vacuum [tex]\(c = 3.0 \times 10^8 \, \text{m/s}\)[/tex], we'll use the relation between the speed of light, frequency, and wavelength. The formula is given by:
[tex]\[ \lambda = \frac{c}{f} \][/tex]
where:
- [tex]\(\lambda\)[/tex] is the wavelength in meters,
- [tex]\(c\)[/tex] is the speed of light,
- [tex]\(f\)[/tex] is the frequency of the light.
### Step 1: Identify the known values
- The speed of light, [tex]\(c = 3.0 \times 10^8 \, \text{m/s}\)[/tex],
- The frequency of the yellow light, [tex]\(f = 5.2 \times 10^{14} \, \text{Hz}\)[/tex].
### Step 2: Apply the formula for wavelength
Plug the values of [tex]\(c\)[/tex] and [tex]\(f\)[/tex] into the formula [tex]\(\lambda = \frac{c}{f}\)[/tex]:
[tex]\[ \lambda = \frac{3.0 \times 10^8 \, \text{m/s}}{5.2 \times 10^{14} \, \text{Hz}} \][/tex]
### Step 3: Perform the division
To simplify the division, divide the coefficients and subtract the exponents:
[tex]\[ \lambda = \frac{3.0}{5.2} \times 10^{8 - 14} \][/tex]
[tex]\[ \lambda = 0.576923076923077 \times 10^{-6} \, \text{m} \][/tex]
### Step 4: Convert the wavelength to standard scientific notation
To express this in standard scientific notation, we move the decimal point so that there's only one non-zero digit before the decimal:
[tex]\[ \lambda = 5.769230769230769 \times 10^{-7} \, \text{m} \][/tex]
### Step 5: Round the coefficient
For simplicity, we can round the coefficient to a reasonable number of significant figures (commonly to three significant figures in physics problems):
[tex]\[ \lambda \approx 5.77 \times 10^{-7} \, \text{m} \][/tex]
### Final Answer
Thus, the wavelength of the yellow light is:
[tex]\[ 5.77 \times 10^{-7} \, \text{meters} \][/tex]
[tex]\[ \lambda = \frac{c}{f} \][/tex]
where:
- [tex]\(\lambda\)[/tex] is the wavelength in meters,
- [tex]\(c\)[/tex] is the speed of light,
- [tex]\(f\)[/tex] is the frequency of the light.
### Step 1: Identify the known values
- The speed of light, [tex]\(c = 3.0 \times 10^8 \, \text{m/s}\)[/tex],
- The frequency of the yellow light, [tex]\(f = 5.2 \times 10^{14} \, \text{Hz}\)[/tex].
### Step 2: Apply the formula for wavelength
Plug the values of [tex]\(c\)[/tex] and [tex]\(f\)[/tex] into the formula [tex]\(\lambda = \frac{c}{f}\)[/tex]:
[tex]\[ \lambda = \frac{3.0 \times 10^8 \, \text{m/s}}{5.2 \times 10^{14} \, \text{Hz}} \][/tex]
### Step 3: Perform the division
To simplify the division, divide the coefficients and subtract the exponents:
[tex]\[ \lambda = \frac{3.0}{5.2} \times 10^{8 - 14} \][/tex]
[tex]\[ \lambda = 0.576923076923077 \times 10^{-6} \, \text{m} \][/tex]
### Step 4: Convert the wavelength to standard scientific notation
To express this in standard scientific notation, we move the decimal point so that there's only one non-zero digit before the decimal:
[tex]\[ \lambda = 5.769230769230769 \times 10^{-7} \, \text{m} \][/tex]
### Step 5: Round the coefficient
For simplicity, we can round the coefficient to a reasonable number of significant figures (commonly to three significant figures in physics problems):
[tex]\[ \lambda \approx 5.77 \times 10^{-7} \, \text{m} \][/tex]
### Final Answer
Thus, the wavelength of the yellow light is:
[tex]\[ 5.77 \times 10^{-7} \, \text{meters} \][/tex]
