Question

Electrons are accelerated through a potential difference of \(38.0 \mathrm{V} .\) The beam of electrons then passes through a single slit. The width of the central fringe of a diffraction pattern formed on a screen 1.00 m away is \(1.13 \mathrm{mm} .\) What is the width of the slit?

Short answer

The width of the slit is approximately \(3.52 \times 10^{-7}\) meters.

Step-by-step solution

Calculate the Velocity of Electrons

The kinetic energy gained by electrons accelerated through a potential difference is given by \( eV = \frac{1}{2}mv^2 \), where \( e \) is the elementary charge (\( 1.6 \times 10^{-19} \) C), \( V \) is the potential difference (38 V), \( m \) is the electron mass (\( 9.11 \times 10^{-31} \) kg), and \( v \) is the velocity. Solve for \( v \):\[ v = \sqrt{\frac{2eV}{m}} = \sqrt{\frac{2 \times 1.6 \times 10^{-19} \times 38}{9.11 \times 10^{-31}}} \approx 3.66 \times 10^6 \text{ m/s} \]

Calculate the Wavelength of Electrons

Using De Broglie's wavelength formula \( \lambda = \frac{h}{mv} \), where \( h \) is Planck's constant (\( 6.626 \times 10^{-34} \) J·s). Find the wavelength:\[ \lambda = \frac{6.626 \times 10^{-34}}{9.11 \times 10^{-31} \times 3.66 \times 10^6} \approx 1.99 \times 10^{-10} \text{ m} \]

Determine the Width of the Slit

In single-slit diffraction, the width of the central fringe \( 2y \) is approximately \( \frac{2\lambda L}{a} \), where \( L \) is the distance from the slit to the screen (1.00 m), \( \lambda \) is the wavelength, and \( a \) is the slit width. Rearrange to solve for \( a \):\[ 1.13 \times 10^{-3} = \frac{2 \times 1.99 \times 10^{-10} \times 1.00}{a} \]Solve for \( a \):\[ a = \frac{2 \times 1.99 \times 10^{-10} \times 1.00}{1.13 \times 10^{-3}} \approx 3.52 \times 10^{-7} \text{ m} \]

Key concepts

Kinetic Energy Calculation

To calculate the kinetic energy of electrons, we use the relationship between kinetic energy and electric potential energy. When electrons are accelerated through a potential difference, they gain kinetic energy equivalent to the electrical energy acquired. The formula to determine this energy is \( eV = \frac{1}{2}mv^2 \), where:

• \( e \) is the elementary charge \((1.6 \times 10^{-19} \text{ C})\),
• \( V \) represents the potential difference, in this case, 38 Volts,
• \( m \) is the mass of an electron \((9.11 \times 10^{-31} \text{ kg})\), and
• \( v \) is the velocity of the electrons, which is the unknown part we wish to solve.

To find \( v \), rearrange the formula: \[ v = \sqrt{\frac{2eV}{m}} \] Plugging in the values will give us the velocity and subsequently confirm the kinetic energy involved in this process.

De Broglie's Wavelength

The concept of wave-particle duality is beautifully illustrated by De Broglie's wavelength. This principle suggests that particles such as electrons also exhibit wave-like properties. The wavelength, known as De Broglie's wavelength, is determined using the formula: \[ \lambda = \frac{h}{mv} \] Here:

• \( h \) is Planck's constant \((6.626 \times 10^{-34} \text{ J·s})\),
• \( m \) is the mass of the electron \((9.11 \times 10^{-31} \text{ kg})\), and
• \( v \) is the velocity obtained from kinetic energy calculations.

By calculating \( \lambda \), we understand the wave characteristics of electrons that are responsible for phenomena like diffraction.

Single-Slit Diffraction

Single-slit diffraction occurs when waves pass through a narrow slit and spread out, forming a pattern of dark and bright fringes. This phenomenon occurs because different parts of the wave interfere with each other. For electrons behaving as waves, this interference forms a central bright fringe, or maxima, and multiple dark fringes.The width of the central fringe in a diffraction pattern can be approximated using: \[ 2y \approx \frac{2\lambda L}{a} \] where:

• \( 2y \) is the central fringe width (1.13 mm in this problem),
• \( \lambda \) is the electron's wavelength,
• \( L \) is the distance to the screen (1.00 m), and
• \( a \) is the slit width.

Determining \( a \), the slit width, involves rearranging the formula and solving.

Electron Velocity

The velocity of an electron is imperative in both kinetic energy calculation and De Broglie wavelength determination. Once electrons are accelerated through a potential difference, their velocity can be computed using the derived expression for kinetic energy: \[ v = \sqrt{\frac{2eV}{m}} \] This velocity is crucial because:

• It quantifies the electrons' speed after acceleration, providing insight into their kinetic energy.
• It is essential in calculating De Broglie's wavelength, linking both particle and wave characteristics.

Calculated using this expression, it confirms the electrons' high-speed suitability for diffraction experiments.

Central Fringe Width

In diffraction patterns, the width of the central fringe is central to understanding the optics of wave interactions at slits. This width, given by the expression \( 2y \approx \frac{2\lambda L}{a}\), must be accurately computed for precision. In the equation:

• \( 2y \) indicates the observable width of the central bright fringe.
• \( \lambda \) is the wavelength of the electrons determined using De Broglie's hypothesis.
• \( L \) is the distance from the slit to the screen, providing the spatial arrangement context.
• Finally, \( a \) provides the width of the slit, which is the subject of the solution.

Accurate measurement and computation of these components ensure precise characterization of electron wave phenomena like diffraction.