The ratio of de-Broglie wavelength of an a particle and a proton accelerated from rest by the same potential is `(1)/(sqrt(m))`,the value of `m` is

Options

A. `4`
B. `16`
C. `2`
D. `8`

Answer

To solve the problem, we need to find the value of \( m \) given that the ratio of the de Broglie wavelengths of an alpha particle and a proton, both accelerated from rest by the same potential, is \( \frac{1}{\sqrt{m}} \).

### Step-by-Step Solution:

1. **Understand the de Broglie wavelength formula**:
The de Broglie wavelength \( \lambda \) of a particle is given by the formula:
\[ \lambda = \frac{h}{p} \]
where \( h \) is Planck's constant and \( p \) is the momentum of the particle. For a particle accelerated through a potential \( V \), the kinetic energy \( KE \) is given by:
\[ KE = qV \]
where \( q \) is the charge of the particle. The momentum \( p \) can be expressed in terms of mass \( m \) and kinetic energy:
\[ KE = \frac{p^2}{2m} \implies p = \sqrt{2m \cdot KE} = \sqrt{2mqV} \]

2. **Write the de Broglie wavelength for the alpha particle**:
For an alpha particle (which has a mass \( 4m_p \), where \( m_p \) is the mass of a proton) and charge \( 2e \):
\[ \lambda_{\alpha} = \frac{h}{\sqrt{2 \cdot 4m_p \cdot (2eV)}} = \frac{h}{\sqrt{16m_p eV}} = \frac{h}{4\sqrt{m_p eV}} \]

3. **Write the de Broglie wavelength for the proton**:
For a proton (mass \( m_p \) and charge \( e \)):
\[ \lambda_{p} = \frac{h}{\sqrt{2m_p \cdot eV}} \]

4. **Calculate the ratio of the wavelengths**:
Now, we find the ratio of the wavelengths:
\[ \frac{\lambda_{\alpha}}{\lambda_{p}} = \frac{\frac{h}{4\sqrt{m_p eV}}}{\frac{h}{\sqrt{2m_p eV}}} \]
Simplifying this gives:
\[ \frac{\lambda_{\alpha}}{\lambda_{p}} = \frac{1}{4} \cdot \sqrt{\frac{2m_p eV}{m_p eV}} = \frac{1}{4} \cdot \sqrt{2} = \frac{\sqrt{2}}{4} \]

5. **Set the ratio equal to the given expression**:
According to the problem, this ratio is also equal to \( \frac{1}{\sqrt{m}} \):
\[ \frac{\sqrt{2}}{4} = \frac{1}{\sqrt{m}} \]

6. **Cross-multiply to solve for \( m \)**:
Cross-multiplying gives:
\[ \sqrt{2} \cdot \sqrt{m} = 4 \implies \sqrt{m} = \frac{4}{\sqrt{2}} = 2\sqrt{2} \]
Squaring both sides results in:
\[ m = (2\sqrt{2})^2 = 4 \cdot 2 = 8 \]

### Final Answer:
Thus, the value of \( m \) is \( 8 \).