Effect of Accelerating Potential on Characteristic X Ray Wavelengths

To determine the correctness of the two statements and the relationship between them, we analyze them individually:

Analysis of Statement-1:
Characteristic $\text{X}$-rays are produced when high-energy incident electrons collide with target atoms and knock out inner-shell electrons. The vacancies created are then filled by electrons transitioning from higher energy levels to lower ones. The energy of the emitted characteristic photon is given by:
$\Delta E = E_2 - E_1 = \frac{hc}{\lambda}$
where $E_1$ and $E_2$ are the bound energy states of the target atom, $h$ is Planck's constant, $c$ is the speed of light, and $\lambda$ is the characteristic wavelength.

Since the energy levels depend uniquely on the atomic number $Z$ and nuclear shielding of the target material (as described by Moseley's Law, $\sqrt{\nu} \propto (Z - b)$), the wavelengths of the characteristic $\text{X}$-rays are intrinsic to the material of the target. Changing the accelerating potential $V$ alters the kinetic energy of the striking electrons ($K = eV$) and thus affects the intensity of the characteristic lines and the cutoff wavelength of continuous $\text{X}$-rays ($\lambda_{\min} = \frac{hc}{eV}$), but it does not alter the atomic energy levels or the characteristic wavelengths $\lambda$.

Therefore, Statement-1 is True.

Analysis of Statement-2:
When an electron beam strikes a metal target inside an $\text{X}$-ray tube, the majority of the kinetic energy ($\sim 99\%$) is transformed into thermal energy (heat) due to multiple inelastic collisions. Only a small fraction ($\sim 1\%$) of the total kinetic energy is converted into electromagnetic energy in the form of continuous (Bremsstrahlung) and characteristic $\text{X}$-ray photons.

Therefore, Statement-2 is True.

Relationship between Statement-1 and Statement-2:
Statement-2 merely states the conservation of energy and the efficiency mechanism of $\text{X}$-ray production in the target. It does not explain why the characteristic wavelengths remain invariant with accelerating potential. The independence of characteristic wavelengths from the accelerating potential is due to the fixed, quantized atomic energy levels of the target material.

Therefore, Statement-2 is NOT the correct explanation for Statement-1.

Conclusion:

• Statement-1 is True.
• Statement-2 is True.
• Statement-2 is NOT a correct explanation for Statement-1.

This corresponds to Option B.