The ratio of de broglie wavelength of electron accelerated through potentials `V_(1)` and `V_(2)` is 1:2 the ratio of potentials `V_(1) : V_(2)` is

Options

A. `4:1`
B. `1:4`
C. `1:2`
D. `2:1`

Answer

To solve the problem, we need to find the ratio of the potentials \( V_1 \) and \( V_2 \) given that the ratio of the de Broglie wavelengths of electrons accelerated through these potentials is \( 1:2 \).

### Step-by-Step Solution:

1. **Understanding the de Broglie Wavelength**:
The de Broglie wavelength \( \lambda \) of a particle is given by the formula:
\[ \lambda = \frac{h}{p} \]
where \( h \) is Planck's constant and \( p \) is the momentum of the particle.

2. **Relating Momentum to Kinetic Energy**:
The momentum \( p \) of an electron can be expressed in terms of its kinetic energy \( KE \):
\[ KE = \frac{1}{2} mv^2 \quad \text{and} \quad p = mv \]
Therefore, we can express kinetic energy in terms of momentum:
\[ KE = \frac{p^2}{2m} \]

3. **Kinetic Energy from Potential**:
The kinetic energy of an electron accelerated through a potential \( V \) is also given by:
\[ KE = eV \]
where \( e \) is the charge of the electron.

4. **Substituting for Momentum**:
From the kinetic energy expression, we can relate momentum to the potential:
\[ p = \sqrt{2m \cdot KE} = \sqrt{2m \cdot eV} \]

5. **Expressing Wavelengths**:
Now, we can express the wavelengths \( \lambda_1 \) and \( \lambda_2 \) for potentials \( V_1 \) and \( V_2 \):
\[ \lambda_1 = \frac{h}{\sqrt{2m e V_1}} \quad \text{and} \quad \lambda_2 = \frac{h}{\sqrt{2m e V_2}} \]

6. **Setting Up the Ratio**:
Given that \( \frac{\lambda_1}{\lambda_2} = \frac{1}{2} \), we can write:
\[ \frac{\frac{h}{\sqrt{2m e V_1}}}{\frac{h}{\sqrt{2m e V_2}}} = \frac{1}{2} \]
Simplifying this gives:
\[ \frac{\sqrt{2m e V_2}}{\sqrt{2m e V_1}} = \frac{1}{2} \]

7. **Simplifying Further**:
The constants \( \sqrt{2m e} \) cancel out, leading to:
\[ \sqrt{\frac{V_2}{V_1}} = \frac{1}{2} \]

8. **Squaring Both Sides**:
Squaring both sides results in:
\[ \frac{V_2}{V_1} = \frac{1}{4} \]

9. **Finding the Ratio of Potentials**:
Therefore, the ratio of the potentials \( V_1 : V_2 \) can be expressed as:
\[ V_1 : V_2 = 4 : 1 \]

### Final Answer:
The ratio of potentials \( V_1 : V_2 \) is \( 4 : 1 \).

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