Find the $(a)$ maximum frequency,and $(b)$ minimum wavelength of $X$-rays produced by $30 \; kV$ electrons.

(N/A) The maximum frequency of $X$-rays is produced when the entire kinetic energy of the electron is converted into a single photon. The energy of the photon is given by $E = h \nu_{\max} = eV$.
Thus,$\nu_{\max} = \frac{eV}{h}$.
Substituting the values: $\nu_{\max} = \frac{(1.6 \times 10^{-19} \; C) \times (30 \times 10^3 \; V)}{6.63 \times 10^{-34} \; J \cdot s} = 7.24 \times 10^{18} \; Hz$.
$(b)$ The minimum wavelength is related to the maximum frequency by $\lambda_{\min} = \frac{c}{\nu_{\max}}$.
Substituting the values: $\lambda_{\min} = \frac{3 \times 10^8 \; m/s}{7.24 \times 10^{18} \; Hz} \approx 0.0414 \times 10^{-9} \; m = 0.0414 \; nm$.