To determine the energy of a photon given its frequency, we can use Planck's equation:
[tex]\[ E = h \cdot f \][/tex]
where:
- [tex]\( E \)[/tex] is the energy of the photon,
- [tex]\( h \)[/tex] is Planck's constant ([tex]\(6.63 \times 10^{-34} \, \text{J} \cdot \text{s}\)[/tex]),
- [tex]\( f \)[/tex] is the frequency of the photon ([tex]\(3.6 \times 10^{15} \, \text{Hz}\)[/tex]).
Let’s calculate the energy step by step:
1. Identify the given values:
- Planck's constant, [tex]\( h = 6.63 \times 10^{-34} \, \text{J} \cdot \text{s} \)[/tex]
- Frequency, [tex]\( f = 3.6 \times 10^{15} \, \text{Hz} \)[/tex]
2. Substitute the values into Planck's equation:
[tex]\[
E = (6.63 \times 10^{-34} \, \text{J} \cdot \text{s}) \times (3.6 \times 10^{15} \, \text{Hz})
\][/tex]
3. Perform the multiplication:
- First, multiply the numerical values: [tex]\( 6.63 \times 3.6 = 23.868 \)[/tex]
- Next, add the exponents of 10: [tex]\( (-34) + (15) = -19 \)[/tex]
Therefore, the product is:
[tex]\[
E \approx 23.868 \times 10^{-19} \, \text{J}
\][/tex]
4. Adjust the scientific notation:
[tex]\[
23.868 \times 10^{-19} \, \text{J} \approx 2.3868 \times 10^{-18} \, \text{J}
\][/tex]
Thus, the calculated energy of the photon is [tex]\( 2.3868 \times 10^{-18} \, \text{J} \)[/tex].
Comparing this value with the given options:
- [tex]\( 1.8 \times 10^{-49} \, \text{J} \)[/tex]
- [tex]\( 2.4 \times 10^{-10} \, \text{J} \)[/tex]
- [tex]\( 1.8 \times 10^{-18} \, \text{J} \)[/tex]
- [tex]\( 2.4 \times 10^{-18} \, \text{J} \)[/tex]
The value [tex]\( 2.4 \times 10^{-18} \, \text{J} \)[/tex] is very close to our calculated result ([tex]\( 2.3868 \times 10^{-18} \, \text{J} \)[/tex]) and is the closest match among the given choices.
Therefore, the correct choice is:
[tex]\[ \boxed{2.4 \times 10^{-18} \, \text{J}} \][/tex]
