Problem 1 - Physics for Scientists and Engineers with Modern Physics

Question

(1I) The neutrons in a parallel beam, each having kinetic energy \(0.030 \mathrm{eV},\) are directed through two slits 0.60 \(\mathrm{mm}\) apart. How far apart will the interference peaks be on a apart. How far apart will the interference peaks be on a screen 1.0 \(\mathrm{m}\) away? [Hint: First find the wavelength of the neutron.

Step-by-Step Solution

Answer

The separation between the interference peaks is approximately 0.885 micrometers.

Step 1: Find the Wavelength of the Neutrons

First, we need to find the wavelength of the neutrons using their kinetic energy. Given the kinetic energy is 0.030 eV, convert this to joules: \[ 1 ext{ eV} = 1.602 imes 10^{-19} ext{ J} \]Thus, the energy in joules is:\[ E = 0.030 imes 1.602 imes 10^{-19} = 4.806 imes 10^{-21} ext{ J} \]Using the relation between kinetic energy and velocity, \(KE = \frac{1}{2}mv^2\), and the de Broglie wavelength formula \(\lambda = \frac{h}{mv}\), we derive:\[ \lambda = \frac{h}{\sqrt{2mE}} \]Here, \(h = 6.626 \times 10^{-34} \text{ m}^2 \text{ kg/s}\) is Planck's constant and \(m = 1.675 \times 10^{-27} \text{ kg}\) is the mass of a neutron.Substitute the values:\[ \lambda = \frac{6.626 \times 10^{-34}}{\sqrt{2 \cdot 1.675 \times 10^{-27} \cdot 4.806 \times 10^{-21}}} \approx 5.31 \times 10^{-10} \text{ m} \]

Step 2: Apply the Double Slit Formula

Now use the double slit interference formula to find the separation of the interference peaks on the screen. The formula for the position of the maxima (bright fringes) is given by:\[ y_m = \frac{m \lambda L}{d} \]where \( m \) is the order of the maximum, \( \lambda \) is the wavelength, \( L \) is the distance to the screen, and \( d \) is the separation between the slits.Given \( d = 0.60 \times 10^{-3} \text{ m} \) and \( L = 1.0 \text{ m} \),The separation of consecutive interference peaks (\(\Delta y\), the distance between maxima, is:\[ \Delta y = \frac{\lambda L}{d} \]Substitute the values:\[ \Delta y = \frac{5.31 \times 10^{-10} \times 1.0}{0.60 \times 10^{-3}} \approx 8.85 \times 10^{-7} \text{ m} \]

Key Concepts

de Broglie wavelength

The concept of de Broglie wavelength revolutionized our understanding of particles at a quantum level. In essence, it describes the wave-like characteristics of particles, such as electrons and neutrons. The de Broglie wavelength (\(\lambda\)) of a particle is determined by its momentum. Momentum, in turn, is the product of mass and velocity. This concept arises from the idea that every particle can exhibit both wave and particle nature, which is foundational in quantum physics.

To calculate the de Broglie wavelength, we use the formula:

• \(\lambda = \frac{h}{mv}\)

where \(h\) is Planck's constant (\(6.626 \times 10^{-34} \text{ m}^2 \text{ kg/s}\)) and \(mv\) is the momentum of the particle. In cases where the kinetic energy (\(KE\)) is known, we can find the velocity with:

• \(v = \sqrt{\frac{2KE}{m}}\)

This can be substituted back into the de Broglie wavelength formula, simplifying to:

• \(\lambda = \frac{h}{\sqrt{2mKE}}\)

Understanding this concept allows us to predict the behavior of particles under various circumstances, such as their interference patterns in experiments.

double slit experiment

The double slit experiment is one of the most profound experiments illustrating the wave nature of matter. It involves directing particles, like electrons or neutrons, through two closely placed slits to observe interference patterns on a screen placed behind the slits.

When particles pass through the slits, they behave like waves. These waves overlap and create regions of constructive interference (bright fringes) and destructive interference (dark fringes). The positions of these maxima (bright spots) can be predicted using the formula:

• \(y_m = \frac{m \lambda L}{d}\)

Here,

• \(m\) is the order of the maximum (e.g., 0, 1, 2,...),
• \(\lambda\) is the wavelength of the particles,
• \(L\) is the distance to the screen,
• \(d\) is the distance between the two slits.

The difference between consecutive maxima, or interference peaks, is calculated as:

• \(\Delta y = \frac{\lambda L}{d}\)

This result comes from measuring the gaps between the points of constructive interference, which visually demonstrate wave-like behavior of particles.

kinetic energy conversion

In the context of physics, kinetic energy conversion plays a crucial role in explaining how energy translates from one form to another. Specifically, when we analyze particles such as neutrons, their kinetic energy tells us about their motion. This concept is essential to calculate their speed when kinetic energy is known. Kinetic energy (\(KE\)) is given by:

• \(KE = \frac{1}{2}mv^2\)

where \(m\) represents mass and \(v\) denotes velocity.

This formula can be rearranged to find the velocity of a particle if its kinetic energy and mass are known:

• \(v = \sqrt{\frac{2KE}{m}}\)

This conversion is vital in calculating the velocity needed to subsequently determine the de Broglie wavelength of a particle. Thus, kinetic energy conversion is not just a mere energy transformation, but also a window into understanding deeper quantum characteristics like wave-particle duality, helping us bridge classical mechanics with quantum mechanics.