To determine which electromagnetic wave has the least amount of energy, we need to use the relationship between the energy of a wave ([tex]\(E\)[/tex]) and its wavelength ([tex]\(\lambda\)[/tex]). The formula connecting these two quantities is given by:
[tex]\[ E = \frac{hc}{\lambda} \][/tex]
where:
- [tex]\(h\)[/tex] is Planck's constant ([tex]\(6.62607015 \times 10^{-34}\)[/tex] Joule-seconds),
- [tex]\(c\)[/tex] is the speed of light in a vacuum ([tex]\(3 \times 10^8\)[/tex] meters per second),
- [tex]\(\lambda\)[/tex] is the wavelength.
Using this formula, we can calculate the energy for each of the waves A, B, C, and D. Here are the steps:
1. Wave A:
[tex]\[
\lambda_A = 3.0 \times 10^2 \, \text{m}
\][/tex]
[tex]\[
E_A = \frac{6.62607015 \times 10^{-34} \times 3 \times 10^8}{3.0 \times 10^2}
\][/tex]
[tex]\[
E_A = 6.62607015 \times 10^{-28} \, \text{J}
\][/tex]
2. Wave B:
[tex]\[
\lambda_B = 4.0 \times 10^{-6} \, \text{m}
\][/tex]
[tex]\[
E_B = \frac{6.62607015 \times 10^{-34} \times 3 \times 10^8}{4.0 \times 10^{-6}}
\][/tex]
[tex]\[
E_B = 4.9695526125 \times 10^{-20} \, \text{J}
\][/tex]
3. Wave C:
[tex]\[
\lambda_C = 1.2 \times 10^{-12} \, \text{m}
\][/tex]
[tex]\[
E_C = \frac{6.62607015 \times 10^{-34} \times 3 \times 10^8}{1.2 \times 10^{-12}}
\][/tex]
[tex]\[
E_C = 1.6565175375 \times 10^{-13} \, \text{J}
\][/tex]
4. Wave D:
[tex]\[
\lambda_D = 2.0 \times 10^{-9} \, \text{m}
\][/tex]
[tex]\[
E_D = \frac{6.62607015 \times 10^{-34} \times 3 \times 10^8}{2.0 \times 10^{-9}}
\][/tex]
[tex]\[
E_D = 9.939105225 \times 10^{-17} \, \text{J}
\][/tex]
From the calculated energies, we have:
- Energy [tex]\(E_A = 6.62607015 \times 10^{-28} \, \text{J}\)[/tex]
- Energy [tex]\(E_B = 4.9695526125 \times 10^{-20} \, \text{J}\)[/tex]
- Energy [tex]\(E_C = 1.6565175375 \times 10^{-13} \, \text{J}\)[/tex]
- Energy [tex]\(E_D = 9.939105225 \times 10^{-17} \, \text{J}\)[/tex]
Comparing these values, the wave with the least amount of energy is Wave A with energy [tex]\(E_A = 6.62607015 \times 10^{-28} \, \text{J}\)[/tex].
Therefore, the electromagnetic wave with the least amount of energy is Wave A.
