Imagine the following Young’s double-slit experiment using matter rather than light: electrons are accelerated through a potential difference of 18 V, pass through two closely spaced slits separated by a distance d, and create an interference pattern. (a) Using Example 37–11 and Section 34–3 as guides, find the required value for d if the first-order interference fringe is to be produced at an angle of 10°. (b) Given the approximate size of atoms, would it be possible to construct the required two-slit set-up for this experiment?

Verified step by step guidance

1. Step 1: Calculate the de Broglie wavelength of the electrons. The de Broglie wavelength is given by \( \lambda = \frac{h}{p} \), where \( h \) is Planck's constant and \( p \) is the momentum of the electron. The momentum can be related to the kinetic energy \( K \) of the electron, which is given by \( K = eV \), where \( e \) is the charge of the electron and \( V \) is the accelerating potential. Use \( p = \sqrt{2m_e eV} \) to find \( \lambda \).
2. Step 2: Use the condition for constructive interference in a double-slit experiment, which is \( d \sin \theta = m \lambda \), where \( d \) is the slit separation, \( \theta \) is the angle of the interference fringe, \( m \) is the order of the fringe (in this case, \( m = 1 \) for the first-order fringe), and \( \lambda \) is the wavelength. Rearrange this equation to solve for \( d \): \( d = \frac{m \lambda}{\sin \theta} \).
3. Step 3: Substitute the values for \( \lambda \) (calculated in Step 1), \( m = 1 \), and \( \theta = 10^\circ \) into the equation from Step 2 to find the required slit separation \( d \).
4. Step 4: Compare the calculated value of \( d \) to the approximate size of atoms (on the order of \( 10^{-10} \) meters). Discuss whether it is feasible to construct a double-slit setup with such a small separation, considering the limitations of current technology and materials.
5. Step 5: Conclude whether the experiment is practically achievable based on the comparison in Step 4, and provide reasoning for your conclusion.

Key Concepts

Wave-Particle Duality

Wave-particle duality is a fundamental concept in quantum mechanics that describes how particles, such as electrons, exhibit both wave-like and particle-like properties. In the context of the double-slit experiment, electrons can create an interference pattern, which is characteristic of waves, despite being particles. This duality is essential for understanding phenomena like interference and diffraction in quantum systems.

Interference Pattern

An interference pattern occurs when two or more waves overlap and combine, resulting in regions of constructive and destructive interference. In the Young's double-slit experiment, the separation of slits allows waves from each slit to interact, producing bright and dark fringes on a detection screen. The angle at which these fringes appear is influenced by the wavelength of the electrons and the distance between the slits.

Potential Difference and Electron Acceleration

The potential difference, measured in volts, is the energy per unit charge that accelerates electrons in an electric field. When electrons are accelerated through a potential difference of 18 V, they gain kinetic energy, which can be calculated using the equation KE = eV, where e is the charge of the electron. This kinetic energy is crucial for determining the wavelength of the electrons, which directly affects the interference pattern produced in the double-slit experiment.