What will be the ratio of de - broglie wave length of `e^(-)`
accelerated with 400 volt and 100 volt :-

Options

A. `1:2`
B. `2:1`
C. `3:10`
D. `10:3`

Explanation

To find the ratio of the de Broglie wavelengths of an electron accelerated by two different voltages (400 volts and 100 volts), we can follow these steps:

### Step-by-Step Solution:

1. **Understand the De Broglie Wavelength Formula**:
The de Broglie wavelength (λ) of an electron can be expressed in terms of the accelerating potential (V) using the formula:
\[ \lambda = \frac{1.226 \times 10^{-9}}{\sqrt{V}} \]
where V is the accelerating potential in volts.

2. **Identify the Two Accelerating Potentials**:
We have two potentials:
- \( V_1 = 400 \, \text{volts} \)
- \( V_2 = 100 \, \text{volts} \)

3. **Write the Wavelengths for Each Potential**:
Using the formula, we can write:
\[ \lambda_1 = \frac{1.226 \times 10^{-9}}{\sqrt{400}} \]
\[ \lambda_2 = \frac{1.226 \times 10^{-9}}{\sqrt{100}} \]

4. **Calculate the Wavelengths**:
- For \( \lambda_1 \):
\[ \lambda_1 = \frac{1.226 \times 10^{-9}}{20} = 6.13 \times 10^{-11} \, \text{m} \]
- For \( \lambda_2 \):
\[ \lambda_2 = \frac{1.226 \times 10^{-9}}{10} = 1.226 \times 10^{-10} \, \text{m} \]

5. **Find the Ratio of the Wavelengths**:
The ratio of the de Broglie wavelengths \( \frac{\lambda_1}{\lambda_2} \) can be expressed as:
\[ \frac{\lambda_1}{\lambda_2} = \frac{1/\sqrt{V_1}}{1/\sqrt{V_2}} = \frac{\sqrt{V_2}}{\sqrt{V_1}} \]
Substituting the values:
\[ \frac{\lambda_1}{\lambda_2} = \frac{\sqrt{100}}{\sqrt{400}} = \frac{10}{20} = \frac{1}{2} \]

6. **Express the Ratio**:
Therefore, the ratio of the de Broglie wavelengths is:
\[ \lambda_1 : \lambda_2 = 1 : 2 \]

### Final Answer:
The ratio of the de Broglie wavelengths of the electron accelerated with 400 volts and 100 volts is \( 1 : 2 \).