Exercises
Find the (a) maximum frequency, and (b) minimum wavelength of X-rays produced by 30 kV electrons.
Given: Accelerating potential V = 30 kV = 30 × 10³ V
Constants: h = 6.63 × 10⁻³⁴ J s, c = 3 × 10⁸ m/s, e = 1.6 × 10⁻¹⁹ C
When an electron is accelerated through potential V, it gains kinetic energy eV. This energy is entirely converted into photon energy at maximum (shortest wavelength, highest frequency).
(a) Maximum frequency:
eV = h × nu_max
nu_max = eV / h = (1.6 × 10⁻¹⁹ × 30 × 10³) / (6.63 × 10⁻³⁴)
nu_max = (4.8 × 10⁻¹⁵) / (6.63 × 10⁻³⁴)
nu_max = 7.24 × 10¹⁸ Hz
(b) Minimum wavelength:
lambda_min = c / nu_max = hc / (eV)
lambda_min = (6.63 × 10⁻³⁴ × 3 × 10⁸) / (1.6 × 10⁻¹⁹ × 30 × 10³)
lambda_min = (1.989 × 10⁻²⁵) / (4.8 × 10⁻¹⁵)
lambda_min = 4.14 × 10⁻¹¹ m = 0.0414 nm
Answer: (a) nu_max = 7.24 × 10¹⁸ Hz; (b) lambda_min = 4.14 × 10⁻¹¹ m (0.0414 nm)