To solve the problem of finding the ratio of the energies per quanta for two electromagnetic radiations with wave numbers in the ratio of [tex]\(2:3\)[/tex], we need to understand the relationship between wave numbers and energy.
1. Understanding the Relation:
The energy [tex]\(E\)[/tex] of a quantum of electromagnetic radiation is directly proportional to its wave number [tex]\(\tilde{\nu}\)[/tex]. This can be expressed as:
[tex]\[
E \propto \tilde{\nu}
\][/tex]
Hence, if we have two wave numbers [tex]\(\tilde{\nu}_1\)[/tex] and [tex]\(\tilde{\nu}_2\)[/tex] in the ratio [tex]\(2:3\)[/tex], their corresponding energies [tex]\(E_1\)[/tex] and [tex]\(E_2\)[/tex] will also be in the same ratio.
2. Given Ratio:
The given wave numbers are in the ratio [tex]\(2:3\)[/tex]. Let's denote the wave numbers as:
[tex]\[
\tilde{\nu}_1 = 2x \quad \text{and} \quad \tilde{\nu}_2 = 3x
\][/tex]
3. Energy Ratio Calculation:
Since energy is directly proportional to wave number:
[tex]\[
E_1 \propto \tilde{\nu}_1 \quad \text{and} \quad E_2 \propto \tilde{\nu}_2
\][/tex]
We can write the ratio of energies as:
[tex]\[
\frac{E_1}{E_2} = \frac{\tilde{\nu}_1}{\tilde{\nu}_2} = \frac{2x}{3x} = \frac{2}{3}
\][/tex]
4. Conclusion:
The ratio of the energies per quanta for the two radiations is therefore:
[tex]\[
\frac{E_1}{E_2} = \frac{2}{3}
\][/tex]
Upon evaluating the available options:
(A) [tex]\(3:2\)[/tex]
(B) [tex]\(9:4\)[/tex]
(C) [tex]\(4:9\)[/tex]
(D) [tex]\(2:3\)[/tex]
We match our result with option (D):
[tex]\[
\boxed{2:3}
\][/tex]
Therefore, the correct answer is (D) [tex]\(2:3\)[/tex].
