An electron having de-Broglie wavelength $\lambda$ is incident on a target in an $X$-ray tube. The cut-off wavelength of the emitted $X$-ray is:

$\Delta \lambda$ is the difference between the wavelength of the $K_\alpha$ line and the minimum wavelength of the continuous $X$-ray spectrum when the $X$-ray tube is operated at a voltage $V$. If the operating voltage is changed to $V / 3$,then the above difference is $\Delta \lambda^{\prime}$. Then:

Assertion : Soft and hard $X-$ rays differ in frequency as well as velocity.
Reason : The penetrating power of hard $X-$ rays is more than the penetrating power of soft $X-$ rays.

Calculate the wavelength of the $K_{\alpha}$ line for $Z=31$, when $a=5 \times 10^7 \, Hz^{1/2}$ for a characteristic $X$-ray spectrum.

The wavelength of the $K_{\alpha}$ line for an element with atomic number $57$ is $\lambda$. What will be the wavelength of the $K_{\alpha}$ line for an element with atomic number $29$?

The binding energy of the innermost electron in tungsten is $40 \, keV$. The energy of the characteristic $X$-rays emitted is: