Question

A double-slit interference experiment is performed with 2.0-ev photons. The same pair of slits is then used for an experiment with electrons. What is the kinetic energy of the electrons if the interference pattern is the same as for the photons (i.e., the spacing between maxima is the same)?

Short answer

Answer: The kinetic energy of the electrons is approximately 2.43 × 10^{-25} Joules.

Step-by-step solution

Recall the energy-wavelength relation for photons

The energy of a photon is related to its wavelength by the equation: E = h * c / λ where E is the energy, h is the Planck's constant (6.626 × 10^{-34} J·s), c is the speed of light (3 × 10^8 m/s), and λ is the wavelength.

Determine the wavelength of the given photons

We are given the energy of the photons: 2.0 eV. First, we need to convert the energy from electron volts to Joules: E = 2 * 1.602 × 10^{-19} J/eV = 3.204 × 10^{-19} J Now, use the energy-wavelength relation for photons to find the wavelength: λ = h * c / E λ = (6.626 × 10^{-34} J·s) * (3 × 10^8 m/s) / (3.204 × 10^{-19} J) λ ≈ 6.21 × 10^{-10} m

Recall the de Broglie relation

The de Broglie relation is used for particles, such as electrons, and relates their wavelength to their momentum: λ = h / p where λ is the wavelength, h is Planck's constant, and p is the momentum of the particle.

Relate momentum to kinetic energy

We know that the momentum of a particle with mass m and velocity v is given by: p = m * v We also know that kinetic energy (KE) is given by: KE = (1/2) * m * v^2 We want to rewrite the momentum equation in terms of kinetic energy. First, solve the KE equation for v: v = sqrt(2 * KE/m) Now substitute this expression for v in the momentum equation: p = m * sqrt(2 * KE/m)

Find the kinetic energy of the electrons

We have the same wavelength for the electrons as for the photons, λ = 6.21 × 10^{-10} m. Using the de Broglie relation and the expression for momentum in terms of kinetic energy, we get: λ = h / (m * sqrt(2 * KE/m)) Now, the mass of an electron is: m_e = 9.11 × 10^{-31} kg. We can rewrite the equation and solve for KE: KE = h^2 / (2 * m_e * λ^2) Plug in the values of h, m_e, and λ to find the kinetic energy of the electrons: KE = (6.626 × 10^{-34} J·s)^2 / (2 * (9.11 × 10^{-31} kg) * (6.21 × 10^{-10} m)^2) KE ≈ 2.43 × 10^{-25} J So, the kinetic energy of the electrons is about 2.43 × 10^{-25} Joules.