To find the frequency and energy of light based on its wavelength, we can use the following key physics formulas:
1. The speed of light formula to find frequency ([tex]\( f \)[/tex]):
[tex]\[
f = \frac{c}{\lambda}
\][/tex]
where:
- [tex]\( c \)[/tex] is the speed of light ([tex]\( 3 \times 10^8 \)[/tex] meters per second),
- [tex]\( \lambda \)[/tex] is the wavelength of the light.
2. Planck's equation to find energy ([tex]\( E \)[/tex]):
[tex]\[
E = h \times f
\][/tex]
where:
- [tex]\( h \)[/tex] is Planck's constant ([tex]\( 6.626 \times 10^{-34} \)[/tex] joule-seconds),
- [tex]\( f \)[/tex] is the frequency.
Let's calculate the frequency and energy for both types of light:
### Wavelength Conversion:
- Wavelength of red light ([tex]\( \lambda_{\text{red}} \)[/tex]): [tex]\( 750 \)[/tex] nm
[tex]\[
\lambda_{\text{red}} = 750 \times 10^{-9}\ \text{m} = 7.50 \times 10^{-7}\ \text{m}
\][/tex]
- Wavelength of violet light ([tex]\( \lambda_{\text{violet}} \)[/tex]): [tex]\( 400 \)[/tex] nm
[tex]\[
\lambda_{\text{violet}} = 400 \times 10^{-9}\ \text{m} = 4.00 \times 10^{-7}\ \text{m}
\][/tex]
### Frequency Calculation:
- Frequency of red light ([tex]\( f_{\text{red}} \)[/tex]):
[tex]\[
f_{\text{red}} = \frac{c}{\lambda_{\text{red}}} = \frac{3 \times 10^8\ \text{m/s}}{7.50 \times 10^{-7}\ \text{m}} = 4.00 \times 10^{14}\ \text{Hz}
\][/tex]
- Frequency of violet light ([tex]\( f_{\text{violet}} \)[/tex]):
[tex]\[
f_{\text{violet}} = \frac{c}{\lambda_{\text{violet}}} = \frac{3 \times 10^8\ \text{m/s}}{4.00 \times 10^{-7}\ \text{m}} = 7.50 \times 10^{14}\ \text{Hz}
\][/tex]
### Energy Calculation:
- Energy of red light ([tex]\( E_{\text{red}} \)[/tex]):
[tex]\[
E_{\text{red}} = h \times f_{\text{red}} = 6.626 \times 10^{-34}\ \text{J} \cdot \text{s} \times 4.00 \times 10^{14}\ \text{Hz} = 2.6504 \times 10^{-19}\ \text{J}
\][/tex]
- Energy of violet light ([tex]\( E_{\text{violet}} \)[/tex]):
[tex]\[
E_{\text{violet}} = h \times f_{\text{violet}} = 6.626 \times 10^{-34}\ \text{J} \cdot \text{s} \times 7.50 \times 10^{14}\ \text{Hz} = 4.9695 \times 10^{-19}\ \text{J}
\][/tex]
### Summary:
- Red light of 750 nm:
- Frequency: [tex]\( 4.00 \times 10^{14} \)[/tex] Hz
- Energy: [tex]\( 2.6504 \times 10^{-19} \)[/tex] J
- Violet light of 400 nm:
- Frequency: [tex]\( 7.50 \times 10^{14} \)[/tex] Hz
- Energy: [tex]\( 4.9695 \times 10^{-19} \)[/tex] J
