A beam of electrons is accelerated from rest through a potential difference of 0.200 kV and then passes through a thin slit. When viewed far

Question

A beam of electrons is accelerated from rest through a potential difference of 0.200 kV and then passes through a thin slit. When viewed far from the slit, the diffracted beam shows its first diffraction minima at ± 13.6 ∘ from the original direction of the beam.

Do we need to use relativity formulas? Select the correct answer and explanation.

a. No. The electrons gain kinetic energy K as they are accelerated through a potential difference V, so Ve=K=mc2/(γ−1). The potential difference is 0.200 kV , soVe= 0.200 keV. Solving for γ and using the fact that the rest energy of an electron is0.511 MeV, we have γ–1=(0.511MeV)/(0.200keV) so γ−1>>1 which means that we do not have to use special relativity.
b. Yes. The electrons gain kinetic energy K as they are accelerated through a potential difference V, so Ve=K=(γ−1)mc2. The potential difference is 0.200 kV , soVe= 0.200 keV. Solving for γ and using the fact that the rest energy of an electron is0.511 MeV, we have γ–1=(0.200keV)/(0.511MeV) so γ<<1 which means that we have to use special relativity.
c. Yes. The electrons gain kinetic energy K as they are accelerated through a potential difference V, so Ve=K=mc2/(γ−1). The potential difference is 0.200 kV , soVe= 0.200 keV. Solving for γ and using the fact that the rest energy of an electron is0.511 MeV, we have γ–1=(0.511MeV)/(0.200keV) so γ>>1 which means that we have to use special relativity.
d. No. The electrons gain kinetic energy K as they are accelerated through a potential difference V, so Ve=K=(γ−1)mc2. The potential difference is 0.200 kV , soVe= 0.200 keV. Solving for γ and using the fact that the rest energy of an electron is 0.511 MeV, we have γ–1=(0.200keV)/(0.511MeV) so γ−1<<1 which means that we do not have to use special relativity.

Part B

How wide is the slit?

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Đan Thu 4 years 2021-09-03T18:07:41+00:00 1 Answers 109 views 0

Answers ( )

  1. Orla
    0
    2021-09-03T18:08:58+00:00
    Reply

    Answer:

    a) the correct answer is d
    , b)    a = 3.69 10⁻⁸ m

    Explanation:

    a) to see which answer is correct let’s sketch the solution to the problem

                ΔU = K

                K = (γ -1) mc²

                (γ -1) = K / mc²

                (γ-1) = ΔU / mc²

                (γ-1) = e V / mc²

               

    If we work in electron volt units ΔU = V    [eV]

               (γ-1) = V / mc²

               (γ -1) = 0.2 10³ / 0.511 10⁶

               (γ -1) = 3.9 10⁻⁴

    As it is very small, relativistic corrections are not necessary.

    Checking the correct answer is d

    b) let’s use De Broglie’s relationship to find the wavelength of electrons

               λ = h / p = h / mv

    Let’s look for the speed of electrons, for this we use the concept of energy conservation

    Initial

            Em₀ = ΔU = e ΔV

    Final

             Emf = K = ½ m v²

             Em₀ = Emf

            e ΔV = ½ m v²

            v = √ (2 e ΔV / m)

            v = √ (2 1.60 10⁻¹⁹ 0.2 10³ / 9.1 10⁻³¹)

            v = √ (70.33 10¹²)

            v = 8.39 10⁶ m / s

    Much less than the speed of light

    We replace

                 λ = 6.63 10⁻³⁴ / (9.1 10⁻³¹ 8.39 10⁶)

                λ = 8.68 10⁻⁹ m

    The diffraction is explained by the expression

               a sin θ = m λ

    The minimum occurs for m = 1

               a = λ / sin θ

               a = 8.68 10⁻⁹ / sin 13.6

               a = 3.69 10⁻⁸ m

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