Moving with the same velocity . One of the following has the longest deBroglie wavelength

Options

A. `beta` - particle
B. `alpha` - particle
C. Proton
D. Neutron

Answer

To solve the problem of determining which particle has the longest de Broglie wavelength when moving with the same velocity, we can follow these steps:

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

### Step 2: Analyze the relationship
From the formula, we can see that the de Broglie wavelength is inversely proportional to the mass of the particle when the velocity is constant:
\[ \lambda \propto \frac{1}{m} \]
This means that as the mass (\( m \)) decreases, the wavelength (\( \lambda \)) increases.

### Step 3: Compare the masses of the particles
We need to compare the masses of the given particles:
- **Beta particle**: This is an electron or a positron, which has a very small mass (approximately \( 9.11 \times 10^{-31} \) kg).
- **Alpha particle**: This consists of 2 protons and 2 neutrons, with a mass of about \( 6.64 \times 10^{-27} \) kg.
- **Proton**: The mass of a proton is approximately \( 1.67 \times 10^{-27} \) kg.
- **Neutron**: The mass of a neutron is approximately \( 1.68 \times 10^{-27} \) kg.

### Step 4: Identify the particle with the smallest mass
Among the particles listed:
- The beta particle has the smallest mass, followed by the proton, neutron, and then the alpha particle.

### Step 5: Conclude which particle has the longest de Broglie wavelength
Since the de Broglie wavelength is inversely proportional to mass, the particle with the smallest mass will have the longest wavelength. Therefore, the beta particle will have the longest de Broglie wavelength when all particles are moving with the same velocity.

### Final Answer
The particle with the longest de Broglie wavelength is the **beta particle**.

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