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_neutron = 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 the Planck 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 speed.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.Plugging in the given values, we get:$$d = \frac{(6.626 	imes 10^{-34} 	ext{ J} \cdot 	ext{s})(1 	ext{ m})}{(2.7 	imes 10^{-3} 	ext{ m})(1.675 	imes 10^{-27} 	ext{ kg})(2.0 	imes 10^6 	ext{ m/s})}$$$$d = 1.5 	imes 10^{-6} 	ext{ m} = 1.5 \mu	ext{m}$$Therefore, the correct answer is A.

Electrons Emerge from an Electron Gun with a Speed of 2.0