The ratio of de broglie wavelength of a proton and an alpha particle moving with the same velocity is

Options

A. 1
B. 2
C. 4
D. 0.25

Explanation

To find the ratio of the de Broglie wavelength of a proton and an alpha particle moving with the same velocity, we can follow these steps:

### Step 1: Understand the de Broglie wavelength formula
The de Broglie wavelength (\( \lambda \)) of a particle is given by the formula:
\[ \lambda = \frac{h}{mv} \]
where:
- \( h \) is Planck's constant,
- \( m \) is the mass of the particle,
- \( v \) is the velocity of the particle.

### Step 2: Set up the ratio of wavelengths
Since both the proton and the alpha particle are moving with the same velocity (\( v \)), we can express the ratio of their de Broglie wavelengths as:
\[ \frac{\lambda_{\text{proton}}}{\lambda_{\text{alpha}}} = \frac{h/(m_{\text{proton}} \cdot v)}{h/(m_{\text{alpha}} \cdot v)} \]

### Step 3: Simplify the ratio
The \( h \) and \( v \) cancel out in the ratio, leading to:
\[ \frac{\lambda_{\text{proton}}}{\lambda_{\text{alpha}}} = \frac{m_{\text{alpha}}}{m_{\text{proton}}} \]

### Step 4: Determine the masses
- The mass of a proton (\( m_{\text{proton}} \)) is approximately 1 atomic mass unit (u).
- The mass of an alpha particle (\( m_{\text{alpha}} \)) is approximately 4 atomic mass units (u) since it consists of 2 protons and 2 neutrons.

### Step 5: Substitute the masses into the ratio
Now we can substitute the masses into the ratio:
\[ \frac{\lambda_{\text{proton}}}{\lambda_{\text{alpha}}} = \frac{4 \, \text{u}}{1 \, \text{u}} = 4 \]

### Conclusion
Thus, the ratio of the de Broglie wavelength of a proton to that of an alpha particle is:
\[ \frac{\lambda_{\text{proton}}}{\lambda_{\text{alpha}}} = 4 \]

### Final Answer
The ratio of the de Broglie wavelength of a proton to that of an alpha particle is \( 4:1 \).

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