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 find the ratio of de Broglie wavelengths of an alpha particle and a proton when both have equal momentum, we can follow these steps:

### Step 1: Understand the de Broglie wavelength formula
The de Broglie wavelength (λ) 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: Write the expression for the de Broglie wavelengths of both particles
Let:
- \( \lambda_{\alpha} \) be the de Broglie wavelength of the alpha particle.
- \( \lambda_{p} \) be the de Broglie wavelength of the proton.

Using the formula, we have:
\[ \lambda_{\alpha} = \frac{h}{p_{\alpha}} \]
\[ \lambda_{p} = \frac{h}{p_{p}} \]

### Step 3: Set the momenta equal
Since we are given that the momentum of both particles is equal, we can write:
\[ p_{\alpha} = p_{p} = p \]

### Step 4: Substitute the equal momentum into the wavelength expressions
Now substituting \( p \) into the wavelength expressions:
\[ \lambda_{\alpha} = \frac{h}{p} \]
\[ \lambda_{p} = \frac{h}{p} \]

### Step 5: Find the ratio of the de Broglie wavelengths
Now, we can find the ratio of the de Broglie wavelengths:
\[ \frac{\lambda_{\alpha}}{\lambda_{p}} = \frac{\frac{h}{p}}{\frac{h}{p}} = \frac{h}{p} \times \frac{p}{h} = 1 \]

### Conclusion
Thus, the ratio of the de Broglie wavelengths of the alpha particle to that of the proton when both have equal momentum is:
\[ \frac{\lambda_{\alpha}}{\lambda_{p}} = 1 \]

### Final Answer
The ratio of de Broglie wavelengths is \( 1:1 \).

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