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:
\[ \lambda = \frac{h}{mv} \]
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 problem
We know from the problem statement that:
- The mass ratio of the alpha particle (mₐ) to the proton (mₚ) is given as:
\[ \frac{mₐ}{mₚ} = \frac{4}{1} \]
This means:
\[ mₐ = 4mₚ \]
- Both particles are moving with the same velocity (v).

### Step 3: Write the de-Broglie wavelength for both particles
For the alpha particle:
\[ \lambdaₐ = \frac{h}{mₐ v} \]
For the proton:
\[ \lambdaₚ = \frac{h}{mₚ v} \]

### Step 4: Find the ratio of their wavelengths
To find the ratio of the de-Broglie wavelengths, we compute:
\[ \frac{\lambdaₐ}{\lambdaₚ} = \frac{\frac{h}{mₐ v}}{\frac{h}{mₚ v}} = \frac{mₚ}{mₐ} \]
Since \( v \) and \( h \) are constants and cancel out.

### Step 5: Substitute the mass ratio
Using the mass ratio \( \frac{mₐ}{mₚ} = \frac{4}{1} \), we can express \( \frac{mₚ}{mₐ} \):
\[ \frac{mₚ}{mₐ} = \frac{1}{4} \]
Thus:
\[ \frac{\lambdaₐ}{\lambdaₚ} = \frac{mₚ}{mₐ} = \frac{1}{4} \]

### Step 6: Conclusion
The ratio of the de-Broglie wavelengths of the alpha particle to the proton is:
\[ \frac{\lambdaₐ}{\lambdaₚ} = \frac{1}{4} \]

### Final Answer:
The ratio of their de-Broglie wavelengths is \( 1:4 \).

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