If an alpha particle and a deuteron move with velocity v and 2v respectively, the ratio of their de-Broglie wave length will be ……….

Options

A. `1 : sqrt(2)`
B. `2 : 1`
C. `1 : 1`
D. `sqrt(2) : 1`

Answer

To solve the problem of finding the ratio of the de Broglie wavelengths of an alpha particle and a deuteron moving with velocities \(v\) and \(2v\) respectively, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Masses**:
- The mass of a deuteron (which consists of one proton and one neutron) can be denoted as \(2m\).
- The mass of an alpha particle (which consists of two protons and two neutrons) can be denoted as \(4m\).

2. **Recall the de Broglie Wavelength Formula**:
- The de Broglie wavelength \(\lambda\) is given by the formula:
\[ \lambda = \frac{h}{mv} \]
where \(h\) is Planck's constant, \(m\) is the mass of the particle, and \(v\) is its velocity.

3. **Calculate the de Broglie Wavelength for the Deuteron**:
- For the deuteron moving with velocity \(2v\):
\[ \lambda_d = \frac{h}{(2m)(2v)} = \frac{h}{4mv} \]

4. **Calculate the de Broglie Wavelength for the Alpha Particle**:
- For the alpha particle moving with velocity \(v\):
\[ \lambda_\alpha = \frac{h}{(4m)(v)} = \frac{h}{4mv} \]

5. **Find the Ratio of the de Broglie Wavelengths**:
- To find the ratio of the de Broglie wavelengths \(\frac{\lambda_d}{\lambda_\alpha}\):
\[ \frac{\lambda_d}{\lambda_\alpha} = \frac{\frac{h}{4mv}}{\frac{h}{4mv}} = 1 \]

6. **Conclusion**:
- The ratio of the de Broglie wavelengths of the alpha particle to the deuteron is:
\[ \lambda_d : \lambda_\alpha = 1 : 1 \]

### Final Answer:
The ratio of their de Broglie wavelengths is \(1 : 1\).

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