In the Q-20, find ratio of de-Broglie wavelengths when both have equal momentum

Options

A. `2:1`
B. `1:1`
C. `4:1`
D. `1:4`

Answer

To solve the problem of finding the ratio of de Broglie wavelengths when both a particle and a proton have equal momentum, we can follow these steps:

### Step 1: Understand the de Broglie wavelength formula
The de Broglie wavelength (\( \lambda \)) is given by the formula:
\[ \lambda = \frac{h}{p} \]
where \( h \) is Planck's constant and \( p \) is the momentum of the particle.

### Step 2: Express momentum in terms of mass and velocity
Momentum (\( p \)) can be expressed as:
\[ p = mv \]
where \( m \) is the mass of the particle and \( v \) is its velocity.

### Step 3: Write the de Broglie wavelength in terms of mass and velocity
Substituting the expression for momentum into the de Broglie wavelength formula, we get:
\[ \lambda = \frac{h}{mv} \]

### Step 4: Set the momenta of the particle and proton equal
According to the problem, both the particle and the proton have equal momentum. Therefore, we can write:
\[ p_{\text{particle}} = p_{\text{proton}} \]
This means:
\[ m_{\text{particle}} v_{\text{particle}} = m_{\text{proton}} v_{\text{proton}} \]

### Step 5: Find the ratio of de Broglie wavelengths
Using the de Broglie wavelength formula for both the particle and the proton, we have:
\[ \lambda_{\text{particle}} = \frac{h}{m_{\text{particle}} v_{\text{particle}}} \]
\[ \lambda_{\text{proton}} = \frac{h}{m_{\text{proton}} v_{\text{proton}}} \]

### Step 6: Calculate the ratio of the wavelengths
Now, we can find the ratio of the de Broglie wavelengths:
\[ \frac{\lambda_{\text{particle}}}{\lambda_{\text{proton}}} = \frac{\frac{h}{m_{\text{particle}} v_{\text{particle}}}}{\frac{h}{m_{\text{proton}} v_{\text{proton}}}} = \frac{m_{\text{proton}} v_{\text{proton}}}{m_{\text{particle}} v_{\text{particle}}} \]

### Step 7: Substitute the equal momentum condition
Since \( m_{\text{particle}} v_{\text{particle}} = m_{\text{proton}} v_{\text{proton}} \), we can substitute this into our ratio:
\[ \frac{\lambda_{\text{particle}}}{\lambda_{\text{proton}}} = \frac{m_{\text{proton}} v_{\text{proton}}}{m_{\text{particle}} v_{\text{particle}}} = 1 \]

### Final Answer
Thus, the ratio of the de Broglie wavelengths is:
\[ \frac{\lambda_{\text{particle}}}{\lambda_{\text{proton}}} = 1:1 \]