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)

Question

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. For the electrons, we have:$$\lambda_e = \frac{h}{p_e} = \frac{h}{m_ev_e}$$where m_e is the mass of the electron, and v_e is the speed of the electron. For the neutrons, we have:$$\lambda_n = \frac{h}{p_n} = \frac{h}{m_nv_n}$$where m_n is the mass of the neutron, and v_n is the speed of the neutron. In order for the interference fringes to have the same spacing, we must have:$$\lambda_e = \lambda_n$$Therefore, we have:$$\frac{h}{m_ev_e} = \frac{h}{m_nv_n}$$Solving for v_n, we get:$$v_n = \frac{m_e}{m_n}v_e$$Substituting the given values, we get:$$v_n = \frac{9.11 	imes 10^{-31} 	ext{ kg}}{1.675 	imes 10^{-27} 	ext{ kg}} 	imes 4.0 	imes 10^4 	ext{ m/s} = 2.2 	imes 10^1 	ext{ m/s}$$Therefore, the correct answer is A.

Electrons Are Accelerated to a Speed of 4 10410 ^ { 4 }104

Electrons Are Accelerated to a Speed of 4 $10 ^ { 4 }$