3. X-ray Generation [6 pts] When a high energy beam of electrons is incident on a target material, in addition to the generation of a lot of heat, several physical processes occur: the generation of X-rays and Auger electrons. (a) Calculate the velocity AND kinetic energy (in Joules) with which the electrons strike the target of an xray tube operated at 40 kilovolts. (b) What is the short-wavelength limit of the continuous spectrum (Bremsstralung) emitted when the x-ray tube is operated at 40 kilovolts

a) v = 1,186 10⁸ m / s, K = 6.4 10⁻¹⁵ J b) λ = 3.1265 10⁻¹⁰ m

Explanation:

Electrons are accelerated in the presence of potential, we will assume that all this energy is converted into kinetic energy of electrons

Em₀ = U = q ΔV

Emf = K = ½ m v²

Em₀ = emf

q ΔV = ½ m v²

v = √2q ΔV / m

Let's calculate

v = √ (2 1.6 10⁻¹⁹ / 9.1 10⁻³¹ 40 10³)

v = √ (1.40659 10¹⁶)

v = 1,186 10⁸ m / s

The kinetic energy is equal to the potential energy

K = q ΔV

K = 1.6 10⁻¹⁹ 40 10³

K = 6.4 10⁻¹⁵ J

b) Let's use the Planck equation

E = h f

c = λ f

E = hc / λ

λ = hc / E

λ = hc / qDV

Let's calculate

λ = 6.67 10⁻³⁴ 3 10⁸ / (1.6 10⁻¹⁹ 40 10³)

λ = 3.1265 10⁻¹⁰ m