A beam of electrons with speed `v_(0)` passes through double slits and then is allowed to strike a fluorescent screen. An interference pattern is observed on the screen.
The electrons are instead replaced by protons moving with speed `v_(0)` Compared to electrons at the same speed, what happens to the spacing of the interference fringes on the screen?

Options

A. The spacing increases
B. The spacing decreases
C. The spacing remains the same
D. None of the above

Answer

To determine the effect on the spacing of the interference fringes when electrons are replaced by protons while maintaining the same speed, we can follow these steps:

### Step 1: Understand the Interference Pattern
The interference pattern created by particles passing through double slits is determined by the wavelength of the particles. The spacing of the interference fringes (Δy) on the screen is given by the formula:

\[ \Delta y = \frac{\lambda L}{d} \]

where:
- \( \Delta y \) = spacing between the interference fringes,
- \( \lambda \) = wavelength of the particle,
- \( L \) = distance from the slits to the screen,
- \( d \) = distance between the slits.

### Step 2: Determine the Wavelength of the Particles
The wavelength (λ) of a particle can be calculated using the de Broglie wavelength formula:

\[ \lambda = \frac{h}{p} \]

where:
- \( h \) = Planck's constant,
- \( p \) = momentum of the particle.

The momentum \( p \) is given by:

\[ p = mv \]

where:
- \( m \) = mass of the particle,
- \( v \) = speed of the particle.

### Step 3: Compare the Wavelengths of Electrons and Protons
For electrons and protons moving with the same speed \( v_0 \):

- The mass of an electron (\( m_e \)) is much smaller than the mass of a proton (\( m_p \)).
- Therefore, the momentum of the electron is \( p_e = m_e v_0 \) and for the proton, it is \( p_p = m_p v_0 \).

Using the de Broglie wavelength formula:

\[ \lambda_e = \frac{h}{m_e v_0} \quad \text{(for electrons)} \]
\[ \lambda_p = \frac{h}{m_p v_0} \quad \text{(for protons)} \]

Since \( m_p > m_e \), it follows that:

\[ \lambda_p < \lambda_e \]

### Step 4: Effect on the Spacing of the Interference Fringes
Since the wavelength of the protons is less than that of the electrons, we can substitute this into the fringe spacing formula:

\[ \Delta y_p = \frac{\lambda_p L}{d} < \frac{\lambda_e L}{d} = \Delta y_e \]

Thus, the spacing of the interference fringes for protons will be less than that for electrons.

### Conclusion
When protons replace electrons at the same speed, the spacing of the interference fringes on the screen decreases.

### Final Answer
The spacing of the interference fringes decreases when protons replace electrons at the same speed.

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