Question 40.45: X rays are produced in an x-ray tube by electrons accelerate......
X rays are produced in an x-ray tube by electrons accelerated through an electric potential difference of 50.0 kV. Let K_0 be the kinetic energy of an electron at the end of the acceleration. The electron collides with a target nucleus (assume the nucleus remains stationary) and then has kinetic energy K_1 = 0.500K_0. (a) What wavelength is associated with the photon that is emitted? The electron collides with another target nucleus (assume it, too, remains stationary) and then has kinetic energy K_2 = 0.500K_1. (b) What wavelength is associated with the photon that is emitted?
The initial kinetic energy of the electron is K_0 = 50.0 keV. After the first collision, the kinetic energy is K_1 = 25 keV; after the second, it is K_2 = 12.5 keV; and after the third, it is zero.
(a) The energy of the photon produced in the first collision is 50.0 keV – 25.0 keV = 25.0 keV. The wavelength associated with this photon is
\lambda=\frac{h c}{E}=\frac{1240 \,eV \cdot nm }{25.0 \times 10^3 \,eV }=4.96 \times 10^{-2} \,nm =49.6 \,pm
where we have used hc = 1240 eV·nm.
(b) The energies of the photons produced in the second and third collisions are each 12.5 keV and their wavelengths are
\lambda=\frac{1240 \,eV \cdot nm }{12.5 \times 10^3 \,eV }=9.92 \times 10^{-2} \,nm =99.2 \,pm .