An electron of a velocity 'x' is found to have a certain wavelength. The velocity to be possessed by the neutron to have half the de Broglie wavelength possessed by electron is:

Options

A. `x//1840`
B. `x//920`
C. `3680x`
D. `x//3680`

Explanation

To solve the problem, we need to calculate the velocity that a neutron must have to achieve half the de Broglie wavelength of an electron moving at velocity 'x'.

### Step-by-Step Solution:

1. **Understand the de Broglie wavelength formula**:
The de Broglie wavelength (λ) of a particle is given by the formula:
\[ \lambda = \frac{h}{mv} \]
where \( h \) is Planck's constant, \( m \) is the mass of the particle, and \( v \) is its velocity.

2. **Calculate the wavelength of the electron**:
For the electron with velocity \( x \):
\[ \lambda_e = \frac{h}{m_e \cdot x} \]
where \( m_e \) is the mass of the electron.

3. **Determine the desired wavelength for the neutron**:
We need the neutron to have half the de Broglie wavelength of the electron:
\[ \lambda_n = \frac{1}{2} \lambda_e = \frac{1}{2} \cdot \frac{h}{m_e \cdot x} = \frac{h}{2m_e \cdot x} \]

4. **Set up the equation for the neutron's wavelength**:
For the neutron, the de Broglie wavelength is given by:
\[ \lambda_n = \frac{h}{m_n \cdot v_n} \]
where \( m_n \) is the mass of the neutron and \( v_n \) is the velocity of the neutron.

5. **Equate the two expressions for the wavelength**:
Setting the two expressions for \( \lambda_n \) equal gives:
\[ \frac{h}{m_n \cdot v_n} = \frac{h}{2m_e \cdot x} \]

6. **Cancel \( h \) from both sides**:
\[ \frac{1}{m_n \cdot v_n} = \frac{1}{2m_e \cdot x} \]

7. **Cross-multiply to solve for \( v_n \)**:
\[ 2m_e \cdot x = m_n \cdot v_n \]
Thus,
\[ v_n = \frac{2m_e \cdot x}{m_n} \]

8. **Substitute the known values**:
Using the known masses of the electron and neutron:
- Mass of electron, \( m_e \approx 9.11 \times 10^{-31} \) kg
- Mass of neutron, \( m_n \approx 1.675 \times 10^{-27} \) kg

Therefore, substituting these values gives:
\[ v_n = \frac{2 \cdot (9.11 \times 10^{-31}) \cdot x}{1.675 \times 10^{-27}} \]

9. **Simplify the expression**:
\[ v_n = \frac{18.22 \times 10^{-31} \cdot x}{1.675 \times 10^{-27}} \approx \frac{x}{9.0} \]

Thus, the velocity that the neutron must have to achieve half the de Broglie wavelength of the electron is:
\[ v_n \approx \frac{x}{9.0} \]

### Final Answer:
The velocity possessed by the neutron to have half the de Broglie wavelength of the electron is \( \frac{x}{9} \).

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