An electron with velocity `v` is found to have a certain value of de Broglie wavelength .The velocity that the neutron should process to have the same de Broglie wavelength is

Options

A. `( 1840)/(v)`
B. `1840v`
C. `(v )/( 1840)`
D. v

Answer

To solve the problem, we will use the de Broglie wavelength formula, which relates the wavelength (λ) of a particle to its momentum (p). The formula is given by:

\[ \lambda = \frac{h}{mv} \]

Where:
- \( \lambda \) is the de Broglie wavelength,
- \( h \) is Planck's constant,
- \( m \) is the mass of the particle,
- \( v \) is the velocity of the particle.

### Step-by-Step Solution:

1. **Identify the de Broglie wavelength for the electron**:
For an electron with velocity \( v \), the de Broglie wavelength is given by:
\[ \lambda_e = \frac{h}{m_e v} \]
where \( m_e \) is the mass of the electron.

2. **Set the de Broglie wavelength for the neutron**:
We want the neutron to have the same de Broglie wavelength \( \lambda_n \):
\[ \lambda_n = \frac{h}{m_n v_n} \]
where \( m_n \) is the mass of the neutron and \( v_n \) is the velocity of the neutron.

3. **Equate the two wavelengths**:
Since \( \lambda_e = \lambda_n \), we can set the two equations equal to each other:
\[ \frac{h}{m_e v} = \frac{h}{m_n v_n} \]

4. **Cancel \( h \) from both sides**:
Since \( h \) is a constant and appears in both equations, we can cancel it out:
\[ \frac{1}{m_e v} = \frac{1}{m_n v_n} \]

5. **Cross-multiply to find the relationship**:
Cross-multiplying gives us:
\[ m_n v = m_e v_n \]

6. **Solve for the velocity of the neutron**:
Rearranging the equation to solve for \( v_n \):
\[ v_n = \frac{m_n}{m_e} v \]

7. **Substitute the mass ratio**:
The mass of the neutron \( m_n \) is approximately 1840 times the mass of the electron \( m_e \):
\[ v_n = \frac{1840 m_e}{m_e} v = 1840 v \]

### Final Answer:
The velocity that the neutron should possess to have the same de Broglie wavelength as the electron is:
\[ v_n = \frac{v}{1840} \]