Electrons are accelerated to a speed of 4.0 × $10^4$ m/s and are then aimed at a double-slit apparatus, where interference fringes are detected. If the electrons were replaced by neutrons, what speed must neutrons have to produce interference fringes with the same fringe spacing as that observed with the electrons? (h = 6.626 × 10^-34 J ∙ s, m_electron = 9.11 × 10^-31 kg, m_neutron = 1.675 × 10^-27 kg) A) 2.2 × 10¹ m/s B) 7.3 × 10⁷ m/s C) 1.7 × 10⁶ m/s D) 9.3 × 10² m/s

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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.If the electrons and neutrons are to produce interference fringes with the same fringe spacing, then their de Broglie wavelengths must be the same. Therefore, we can set the two equations for the de Broglie wavelength equal to each other and solve for the velocity of the neutrons:$$\frac{h}{p_e} = \frac{h}{p_n}$$$$p_e = p_n$$$$m_ev_e = m_nv_n$$$$v_n = \frac{m_e}{m_n}v_e$$$$v_n = \frac{(9.11 imes 10^{-31} ext{ kg})}{(1.675 imes 10^{-27} ext{ kg})}(4.0 imes 10^4 ext{ m/s})$$$$v_n = 2.2 imes 10^1 ext{ m/s}$$Therefore, the neutrons must have a speed of 2.2 × 10¹ m/s to produce interference fringes with the same fringe spacing as that observed with the electrons.

Electrons Are Accelerated to a Speed of 4 10410^4104 M/s and Are Then Aimed at a Double-Slit Apparatus, Where

Electrons Are Accelerated to a Speed of 4 $10^4$ M/s and Are Then Aimed at a Double-Slit Apparatus, Where