Electrons emerge from an electron gun with a speed of 2.0 × 10⁶ m/s and then pass through a double-slit apparatus.Interference fringes with spacing of 2.7 mm are detected on a screen far from the double slit.What would the fringe spacing be if the electrons were replaced by neutrons with the same speed? (h = 6.626 × 10^-34 J ∙ s,m_electron = 9.11 × 10^-31 kg,m_neutro_n = 1.675 × 10^-27 kg) A)1.5 μm B)4.9 m C)0.93 nm D)1.2 km

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Quizplus · Accepted Answer

The de Broglie wavelength is given by:$$\lambda = \frac{h}{p}$$where h is Planck's constant, and p is the momentum of the particle. The momentum of a particle is given by:$$p = mv$$where m is the mass of the particle, and v is its velocity.Substituting these equations into the equation for the de Broglie wavelength, we get:$$\lambda = \frac{h}{mv}$$The fringe spacing is given by:$$d = \frac{\lambda D}{a}$$where D is the distance between the double slit and the screen, and a is the distance between the slits.Substituting the equation for the de Broglie wavelength into the equation for the fringe spacing, we get:$$d = \frac{hD}{amv}$$The fringe spacing is inversely proportional to the mass of the particle. Therefore, the fringe spacing for neutrons will be smaller than the fringe spacing for electrons.The mass of a neutron is 1.675 × 10^-27 kg, and the mass of an electron is 9.11 × 10^-31 kg. Therefore, the fringe spacing for neutrons will be:$$d = \frac{hD}{amv} = \frac{(6.626 imes 10^{-34} ext{ J s})(1 ext{ m})}{(2.7 imes 10^{-3} ext{ m})(1.675 imes 10^{-27} ext{ kg})(2.0 imes 10^6 ext{ m/s})} = 1.5 imes 10^{-6} ext{ m} = 1.5 \mu ext{m}$$Therefore, the correct answer is A.

Electrons Emerge from an Electron Gun with a Speed of 2.0