A 20 KeV electron emits two bremsstrahlung photons as it is being brought to rest in two successive decelerations. The wavelength of the second photon is 1.30 A longer than the wavelength of the first. (a) What was the energy of the electron after the first deceleration, and (b) what are the wavelengths of the photons

λ₁ = 87.5 10⁻¹² m ,  λ₂ =  2.175 10⁻¹⁰ m,    E₂ = 5.8 10³ eV

Explanation:

In this case you can use the law of conservation of energy, all the energy of the electron is converted into energized emitted photons

Let's reduce to the SI system

          E₀ = 20 10³ eV (1.6 10⁻¹⁹ J / 1eV) = 3.2 10⁻¹⁵ J

          Δλ = 1.30 A = 0.13 nm = 0.13 10⁻⁹ m

          Ef = E₁ + E₂

         E₀ = Ef

         E₀ = E₁ + E₂

The energy can be found with the Planck equation

          E = h f

          c = λ f

          f = c / λ

          E = hc / λ

They indicate that the wavelength of the second photon is

           λ₂ =  λ₁ +0.130 10⁻⁹

We replace

           E₀ = hv / λ₁ + hc / ( λ₁ + 0.130 10⁺⁹)

           E₀ / hv = 1 / λ₁ + 1 / ( λ₁ + 0.13 10⁻⁹)

          3.2 10⁻¹⁵ / (6.63 10⁻³⁴ 3 10⁸) = ( λ₁ + 0.13 10⁻⁹ +  λ₁) /  λ₁ ( λ₁ + 0.13 10⁻⁹)

          1.6 10¹⁰ ( λ₁² +0.13 10⁻⁹  λ₁) = 2  λ₁ + 0.13 10⁻⁹

           λ₁² + 0.13 10⁻⁹  λ₁ = 1.25 10⁻¹⁰  λ₁ + 8.125 10⁻²¹

            λ₁² + 0.005 10⁻⁹  λ₁ = 8.125 10⁻²¹

            λ₁² + 5 10⁻¹²  λ₁ - 8.125 10⁻²¹ = 0

Let's solve the second degree equation

            λ₁ = [-5 10⁻¹² ±√((5 10⁻¹²)² + 4 8.125 10⁻²¹)] / 2

    λ₁ = [-5 10⁻¹² ±√(25 10⁻²⁴ +32.5 10⁻²¹)] / 2 = [-5 10⁻¹² ±√ (32525 10⁻²⁴)] / 2

             λ₁ = [-5 10⁻¹² ± 180 10⁻¹²] / 2

            λ₁ = 87.5 10⁻¹² m

             λ₂ = -92.5 10⁻¹² m

We take the positive wavelength

The wavelength of the photons is

            λ₁ = 87.5 10⁻¹² m

            λ₂ =  λ₁ + 0.13 10⁻⁹

             λ₂ = 87.5 10⁻¹² + 0.13 10⁻⁹

             λ₂ = 0.2175 10⁻⁹ m = 2.175 10⁻¹⁰ m

The energy after the first deceleration is

            E₂ = E₀ –E₁

            E₂ = E₀ –hc / λ₁

            E₂ = 3.2 10⁻¹⁵ - 6.63 10⁺³⁴ 3 10⁸ / 87.5 10⁻¹²

            E₂ = 3.2 10⁻¹⁵ - 2.27 10⁻¹⁵

             E₂ = 0.93 10⁻¹⁵ J

             E₂ = 0.93 10⁻¹⁵ J (1 eV / 1.6 10⁻¹⁹ J)

             E₂ = 5.8 10³ eV

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