If alpha, beta and gamma rays carry same momentum, which has the longest wavelength ?

Options

A. Alpha rays
B. Beta rays
C. Gamma rays
D. None, all have the same wavelength

Answer

To solve the problem of which ray (alpha, beta, or gamma) has the longest wavelength when they all carry the same momentum, we can follow these steps:

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

### Step 2: Analyze the given information
We are given that alpha, beta, and gamma rays all carry the same momentum. This means:
\[ p_{\alpha} = p_{\beta} = p_{\gamma} \]

### Step 3: Apply the de Broglie wavelength formula
Since the momentum is the same for all three types of rays, we can express their wavelengths as follows:
\[ \lambda_{\alpha} = \frac{h}{p} \]
\[ \lambda_{\beta} = \frac{h}{p} \]
\[ \lambda_{\gamma} = \frac{h}{p} \]

### Step 4: Compare the wavelengths
From the equations above, we can see that:
\[ \lambda_{\alpha} = \lambda_{\beta} = \lambda_{\gamma} \]
This means that all three types of rays have the same wavelength when they carry the same momentum.

### Step 5: Conclusion
Since all three rays (alpha, beta, and gamma) have the same wavelength, none of them has a longer wavelength than the others. Therefore, the answer is that they all have the same wavelength.

### Final Answer
All three rays (alpha, beta, and gamma) have the same wavelength.

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