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`

Answer

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

### Step-by-Step Solution:

1. **Understanding the Wavelength Ratio**:
We are given that the wavelengths of electron waves in two orbits are in the ratio of 3:5. Let's denote the wavelengths as:
\[ \lambda_1 = 3x \quad \text{and} \quad \lambda_2 = 5x \]

2. **Using the Wavelength Formula**:
The wavelength of an electron can be expressed using the formula:
\[ \lambda = \frac{h}{\sqrt{2mk}} \]
where \( h \) is Planck's constant, \( m \) is the mass of the electron, and \( k \) is the kinetic energy of the electron.

3. **Setting Up the Proportionality**:
Since the wavelength is inversely proportional to the square root of the kinetic energy, we can write:
\[ \frac{\lambda_1}{\lambda_2} = \frac{\sqrt{k_2}}{\sqrt{k_1}} \]

4. **Substituting the Wavelengths**:
Substituting the values of \( \lambda_1 \) and \( \lambda_2 \):
\[ \frac{3x}{5x} = \frac{\sqrt{k_2}}{\sqrt{k_1}} \]
Simplifying this gives:
\[ \frac{3}{5} = \frac{\sqrt{k_2}}{\sqrt{k_1}} \]

5. **Squaring Both Sides**:
To eliminate the square roots, we square both sides:
\[ \left(\frac{3}{5}\right)^2 = \frac{k_2}{k_1} \]
This results in:
\[ \frac{9}{25} = \frac{k_2}{k_1} \]

6. **Finding the Ratio of Kinetic Energies**:
We need to find the ratio \( \frac{k_1}{k_2} \):
\[ \frac{k_1}{k_2} = \frac{25}{9} \]
Thus, the ratio of the kinetic energies \( k_1 : k_2 \) is:
\[ k_1 : k_2 = 25 : 9 \]

7. **Conclusion**:
Therefore, the ratio of the kinetic energy of electrons in the two orbits is \( 25 : 9 \).