Analysis of Statement 1

1. Production of Characteristic X-rays: Characteristic X-rays are produced when a high-energy electron, accelerated by a potential difference in an X-ray tube, collides with a target atom and ejects an electron from one of the inner shells (e.g., K, L, M shells).
2. Electron Transition: This creates a vacancy in the inner shell. To fill this vacancy, an electron from a higher energy outer shell jumps down to the lower energy inner shell.
3. Photon Emission: The difference in energy between the higher and lower energy shells is released as a photon. If this photon's energy is in the X-ray range, it is called a characteristic X-ray.
4. Energy and Wavelength: The energy of the emitted photon is given by the difference in the energy levels of the atomic shells involved, i.e., $E_{photon} = \Delta E = E_{initial} - E_{final}$. The corresponding wavelength is $\lambda = \frac{hc}{E_{photon}} = \frac{hc}{\Delta E}$.
5. Dependence on Accelerating Potential: The energy levels ($E_{initial}$, $E_{final}$) are intrinsic properties of the atoms of the target material. They depend on the atomic number (Z) of the target element, but they do not depend on the kinetic energy of the incident electron.
6. Role of Accelerating Potential: The accelerating potential ($V$) determines the maximum kinetic energy of the incident electrons ($KE_{max} = eV$). This energy must be greater than the binding energy of the inner-shell electron to be able to eject it. Increasing the accelerating potential increases the kinetic energy of the incident electrons, which increases the intensity of the characteristic X-rays (more ionizations occur) and may allow for the creation of new characteristic lines if new shells can be ionized. However, it does not change the energy difference $\Delta E$ between the fixed atomic shells.
7. Conclusion for Statement 1: Since the wavelengths of characteristic X-rays depend only on the fixed energy level differences of the target atom, they do not change when the accelerating potential is increased (as long as it's above the threshold potential for producing them). Therefore, Statement 1 is True.

Analysis of Statement 2

1. Energy Conversion in an X-ray Tube: In an X-ray tube, electrons are accelerated, gaining kinetic energy. When this high-energy electron beam strikes the metal target, the electrons are rapidly decelerated.
2. X-ray Production Mechanisms: The kinetic energy of the electrons is converted into other forms of energy. A very small fraction (typically less than 1%) is converted into X-ray energy through two main processes:
   • Bremsstrahlung (Continuous X-rays): The deceleration of electrons in the electric field of the target nuclei produces a continuous spectrum of X-rays.
   • Characteristic X-rays: The process described in the analysis of Statement 1.
3. Energy Conservation: In both processes, a portion of the incident electron's kinetic energy is converted into the energy of an X-ray photon. The majority of the kinetic energy is converted into heat in the target.
4. Conclusion for Statement 2: The statement correctly describes the fundamental principle of X-ray production – the conversion of the kinetic energy of electrons into X-ray energy. Therefore, Statement 2 is True.

Analysis of the Relationship between Statement 1 and Statement 2

1. Statement 1 describes a specific property of characteristic X-rays: their wavelengths are independent of the accelerating potential.
2. Statement 2 is a general statement about the energy conversion process that leads to the production of all X-rays (both continuous and characteristic).
3. To be a correct explanation, Statement 2 should explain why the wavelengths of characteristic X-rays are independent of the accelerating potential. The actual reason is that these wavelengths are determined by the discrete and fixed energy levels of the target atoms.
4. Statement 2 does not mention these atomic energy levels. It is a general principle that also applies to the continuous X-ray spectrum. The minimum wavelength of the continuous spectrum ($\") is *directly* dependent on the accelerating potential ($...$). Since Statement 2 applies to a phenomenon that does depend on the accelerating potential, it cannot be the specific reason why another phenomenon (characteristic X-rays) does not.
5. Therefore, while both statements are true, Statement 2 does not provide the specific physical reason for Statement 1. It is not the correct explanation.

Final Conclusion

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

This corresponds to option B.