An a particle and a proton have their masses in the ratio 4:1 and charges in the ratio 2: 1. Find ratio of their de-Broglie wavelengths when both move with equal velocities.

Options

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

Answer

To find the ratio of the de-Broglie wavelengths of an alpha particle and a proton when both are moving with equal velocities, we can follow these steps:

### Step 1: Understand the de-Broglie wavelength formula
The de-Broglie wavelength (λ) is given by the formula:
\[ \lambda = \frac{h}{p} \]
where \(h\) is Planck's constant and \(p\) is the momentum of the particle.

### Step 2: Relate momentum to mass and velocity
The momentum \(p\) can be expressed as:
\[ p = mv \]
where \(m\) is the mass of the particle and \(v\) is its velocity. Therefore, we can rewrite the de-Broglie wavelength as:
\[ \lambda = \frac{h}{mv} \]

### Step 3: Set up the ratio of the wavelengths
Since both particles (the alpha particle and the proton) are moving with equal velocities, we can express the ratio of their wavelengths as:
\[ \frac{\lambda_{\text{alpha}}}{\lambda_{\text{proton}}} = \frac{m_{\text{proton}}}{m_{\text{alpha}}} \]

### Step 4: Use the given mass ratio
From the problem, we know the mass ratio of the alpha particle to the proton is:
\[ \frac{m_{\text{alpha}}}{m_{\text{proton}}} = \frac{4}{1} \]
This implies:
\[ \frac{m_{\text{proton}}}{m_{\text{alpha}}} = \frac{1}{4} \]

### Step 5: Substitute the mass ratio into the wavelength ratio
Now substituting the mass ratio into the wavelength ratio gives:
\[ \frac{\lambda_{\text{alpha}}}{\lambda_{\text{proton}}} = \frac{1}{4} \]

### Step 6: Find the ratio of the wavelengths
Thus, we can conclude that:
\[ \lambda_{\text{alpha}} : \lambda_{\text{proton}} = 1 : 4 \]

### Final Answer
The ratio of the de-Broglie wavelengths of the alpha particle to the proton is:
\[ \lambda_{\text{alpha}} : \lambda_{\text{proton}} = 1 : 4 \]

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