The wavelengths of electron waves in two orbits is 3:5. The ratio of kinetic energy of electrons will be

Options

A. `25:9`
B. `5:3`
C. `9:25`
D. `3:5`

Explanation

To solve the problem of finding the ratio of kinetic energy of electrons in two orbits given the ratio of their wavelengths, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the relationship between wavelength and kinetic energy**:
The de Broglie wavelength (\( \lambda \)) of an electron is given by the formula:
\[ \lambda = \frac{h}{\sqrt{2m \cdot KE}} \]
where \( h \) is Planck's constant, \( m \) is the mass of the electron, and \( KE \) is the kinetic energy of the electron.

2. **Set up the ratio of wavelengths**:
Given that the ratio of the wavelengths in two orbits is:
\[ \frac{\lambda_1}{\lambda_2} = \frac{3}{5} \]

3. **Express the wavelengths in terms of kinetic energy**:
From the wavelength formula, we can express the wavelengths for both orbits:
\[ \lambda_1 = \frac{h}{\sqrt{2m \cdot KE_1}} \quad \text{and} \quad \lambda_2 = \frac{h}{\sqrt{2m \cdot KE_2}} \]

4. **Form the ratio of the wavelengths**:
Taking the ratio of the two wavelengths gives:
\[ \frac{\lambda_1}{\lambda_2} = \frac{\sqrt{2m \cdot KE_2}}{\sqrt{2m \cdot KE_1}} = \sqrt{\frac{KE_2}{KE_1}} \]

5. **Substituting the wavelength ratio**:
We can substitute the known wavelength ratio into the equation:
\[ \frac{3}{5} = \sqrt{\frac{KE_2}{KE_1}} \]

6. **Square both sides**:
Squaring both sides to eliminate the square root gives:
\[ \left(\frac{3}{5}\right)^2 = \frac{KE_2}{KE_1} \]
\[ \frac{9}{25} = \frac{KE_2}{KE_1} \]

7. **Inverting the ratio**:
Since we need the ratio of \( KE_1 \) to \( KE_2 \), we can invert the ratio:
\[ \frac{KE_1}{KE_2} = \frac{25}{9} \]

8. **Final answer**:
Therefore, the ratio of the kinetic energy of electrons in the two orbits is:
\[ KE_1 : KE_2 = 25 : 9 \]

### Final Result:
The ratio of kinetic energy of electrons in the two orbits is \( 25 : 9 \).