In order to have the same wavelength for the electron (mass `m_(e)`) and the neutron (mass `m_(n)`) their velocities should be in the ratio (electron veloctiy/neutron veloctity) :-

Options

A. `m_(n)//m_(e)`
B. `m_(n)xxm_(e)`
C. `m_(e)//m_(n)`
D. one

Answer

To find the ratio of the velocities of an electron and a neutron such that they have the same wavelength, we can follow these steps:

### Step 1: Write the de Broglie wavelength equations
The de Broglie wavelength (\( \lambda \)) for 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.

### Step 2: Set up the equations for the electron and neutron
For the electron, the wavelength (\( \lambda_e \)) is:
\[ \lambda_e = \frac{h}{m_e v_e} \]
For the neutron, the wavelength (\( \lambda_n \)) is:
\[ \lambda_n = \frac{h}{m_n v_n} \]

### Step 3: Set the wavelengths equal
Since we want the wavelengths to be equal (\( \lambda_e = \lambda_n \)), we can set the two equations equal to each other:
\[ \frac{h}{m_e v_e} = \frac{h}{m_n v_n} \]

### Step 4: Cancel out Planck's constant
We can cancel \( h \) from both sides of the equation:
\[ \frac{1}{m_e v_e} = \frac{1}{m_n v_n} \]

### Step 5: Rearrange the equation
Rearranging gives us:
\[ m_n v_n = m_e v_e \]

### Step 6: Express the ratio of velocities
Now, we can express the ratio of the velocities:
\[ \frac{v_e}{v_n} = \frac{m_n}{m_e} \]

### Conclusion
Thus, the ratio of the velocity of the electron to the velocity of the neutron is:
\[ \frac{v_e}{v_n} = \frac{m_n}{m_e} \]