The frequencies of $K_{\alpha}$,$K_{\beta}$,and $L_{\alpha}$ $X$-rays of a material are $\nu_1$,$\nu_2$,and $\nu_3$ respectively. Which of the following relations holds good?

The minimum wavelength of $X$-rays produced by an electron accelerated through a potential difference of $V$ volts is proportional to:

An $X$-ray tube operates at $50 \, kV$ and $20 \, mA$. The target material has a mass of $1.0 \, kg$ and a specific heat capacity of $495 \, J \cdot kg^{-1} \cdot ^\circ C^{-1}$. One percent of the applied electrical power is converted into $X$-rays,and the remaining energy is used to heat the target. Then:
$(1)$ $A$ suitable target material should have a high melting point.
$(2)$ $A$ suitable target material should have low thermal conductivity.
$(3)$ The rate of rise of temperature of the target is $2 \, ^\circ C/s$.
$(4)$ The minimum wavelength of the emitted $X$-rays is $0.25 \, \mathring{A}$.

The wavelength of the series limit of the Lyman series in the hydrogen spectrum is $911 \ \mathring{A}$. Calculate the atomic number of the element that emits characteristic $X$-rays with a minimum wavelength of $0.7 \ \mathring{A}$.

When $0.50\; \mathring{A}$ $X$-rays strike a material,the photoelectrons from the $K$-shell are observed to move in a circle of radius $23\; mm$ in a magnetic field of $2 \times 10^{-2}\; T$ acting perpendicularly to the direction of emission of photoelectrons. The binding energy of $K$-shell electrons is $..........\; keV$.

According to Moseley's law,the frequency of a spectral line in the $X$-ray spectrum varies as: