The ratio of velocities of a proton and an `alpha` particle is 4 : 1. The ratio of their De Broglie wave lengths will be

Options

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

Answer

To solve the problem, we need to find the ratio of the De Broglie wavelengths of a proton and an alpha particle given the ratio of their velocities.

### Step-by-Step Solution:

1. **Understand the De Broglie Wavelength Formula**:
The De Broglie wavelength (λ) 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.

2. **Write the Expressions for the Wavelengths**:
For a proton (p) and an alpha particle (α), the De Broglie wavelengths can be expressed as:
\[ \lambda_p = \frac{h}{m_p v_p} \]
\[ \lambda_\alpha = \frac{h}{m_\alpha v_\alpha} \]

3. **Find the Ratio of the Wavelengths**:
To find the ratio of the wavelengths, we divide the wavelength of the proton by the wavelength of the alpha particle:
\[ \frac{\lambda_p}{\lambda_\alpha} = \frac{\frac{h}{m_p v_p}}{\frac{h}{m_\alpha v_\alpha}} = \frac{m_\alpha v_\alpha}{m_p v_p} \]
Here, \( h \) cancels out.

4. **Substitute the Known Mass and Velocity Ratios**:
We know from the problem that:
- The mass of an alpha particle \( m_\alpha \) is approximately 4 times the mass of a proton \( m_p \):
\[ m_\alpha = 4m_p \]
- The ratio of the velocities is given as \( v_p : v_\alpha = 4 : 1 \), which means:
\[ \frac{v_p}{v_\alpha} = 4 \quad \Rightarrow \quad \frac{v_\alpha}{v_p} = \frac{1}{4} \]

5. **Plug in the Ratios**:
Now, substituting these ratios into the wavelength ratio:
\[ \frac{\lambda_p}{\lambda_\alpha} = \frac{(4m_p)(\frac{1}{4}v_p)}{m_p v_p} \]
Simplifying this gives:
\[ \frac{\lambda_p}{\lambda_\alpha} = \frac{4m_p \cdot \frac{1}{4}v_p}{m_p v_p} = \frac{4 \cdot \frac{1}{4}}{1} = 1 \]

6. **Conclusion**:
Therefore, the ratio of the De Broglie wavelengths of the proton and the alpha particle is:
\[ \lambda_p : \lambda_\alpha = 1 : 1 \]

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