A proton and an `alpha` - particle are accelerated through same potential difference. Then, the ratio of de-Broglie wavelength of proton and `alpha`-particle is

Options

A. `sqrt2`
B. `(1)/(sqrt2)`
C. `2sqrt2`
D. None of these

Explanation

To find the ratio of the de Broglie wavelengths of a proton and an alpha particle when both are accelerated through the same potential difference, we can follow these steps:

### Step 1: Understand the de Broglie wavelength formula
The de Broglie wavelength (λ) of a particle can be expressed as:
\[ \lambda = \frac{h}{\sqrt{2mqV}} \]
where:
- \( h \) = Planck's constant
- \( m \) = mass of the particle
- \( q \) = charge of the particle
- \( V \) = potential difference

### Step 2: Write the expression for the de Broglie wavelength of the proton
For the proton (denoted as \( p \)):
\[ \lambda_p = \frac{h}{\sqrt{2m_p q_p V}} \]
where:
- \( m_p \) = mass of the proton
- \( q_p \) = charge of the proton

### Step 3: Write the expression for the de Broglie wavelength of the alpha particle
For the alpha particle (denoted as \( \alpha \)):
\[ \lambda_\alpha = \frac{h}{\sqrt{2m_\alpha q_\alpha V}} \]
where:
- \( m_\alpha = 4m_p \) (mass of alpha particle is 4 times that of the proton)
- \( q_\alpha = 2q_p \) (charge of alpha particle is 2 times that of the proton)

Substituting these values into the equation for \( \lambda_\alpha \):
\[ \lambda_\alpha = \frac{h}{\sqrt{2(4m_p)(2q_p)V}} = \frac{h}{\sqrt{16m_p q_p V}} = \frac{h}{4\sqrt{m_p q_p V}} \]

### Step 4: Find the ratio of the de Broglie wavelengths
Now, we can find the ratio of the de Broglie wavelengths of the proton to the alpha particle:
\[ \frac{\lambda_p}{\lambda_\alpha} = \frac{\frac{h}{\sqrt{2m_p q_p V}}}{\frac{h}{4\sqrt{m_p q_p V}}} \]
This simplifies to:
\[ \frac{\lambda_p}{\lambda_\alpha} = \frac{4\sqrt{m_p q_p V}}{\sqrt{2m_p q_p V}} = \frac{4}{\sqrt{2}} = 2\sqrt{2} \]

### Final Answer:
Thus, the ratio of the de Broglie wavelength of the proton to that of the alpha particle is:
\[ \frac{\lambda_p}{\lambda_\alpha} = 2\sqrt{2} \]

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