To find the energy of an electromagnetic wave given its frequency, we need to use the Planck-Einstein relation, which links the energy (E) of a photon to its frequency (ν). The formula is:
[tex]\[ E = h \cdot \nu \][/tex]
where:
- [tex]\(E\)[/tex] is the energy of the electromagnetic wave,
- [tex]\(h\)[/tex] is Planck's constant, approximately [tex]\(6.626 \times 10^{-34}\)[/tex] joule-seconds (J·s),
- [tex]\(\nu\)[/tex] (or frequency) is given as [tex]\(8 \times 10^{12}\)[/tex] Hz.
Let's break down the steps:
1. Identify given values:
- Frequency ([tex]\(\nu\)[/tex]) = [tex]\(8 \times 10^{12}\)[/tex] Hz,
- Planck's constant ([tex]\(h\)[/tex]) = [tex]\(6.626 \times 10^{-34}\)[/tex] J·s.
2. Substitute these values into the formula:
[tex]\[ E = (6.626 \times 10^{-34}\, \text{J·s}) \times (8 \times 10^{12}\, \text{Hz}) \][/tex]
3. Calculate the energy:
[tex]\[ E = 6.626 \times 8 \times 10^{-34 + 12} \, \text{J} \][/tex]
[tex]\[ E = 53.008 \times 10^{-22} \, \text{J} \][/tex]
[tex]\[ E = 5.3008 \times 10^{-21} \, \text{J} \][/tex]
So, the energy of the electromagnetic wave is [tex]\(5.3 \times 10^{-21}\)[/tex] joules.
Among the given options, the correct one is:
C. [tex]\(5.3 \times 10^{-21} J\)[/tex]
