Incorrect Statement Regarding Continuous and Characteristic X-Rays

To determine the incorrect statement, let us analyze each option based on the physics of X-ray production:

1. Analysis of Characteristic X-rays (Option A):
   According to Moseley's law, the frequency $\nu$ of a characteristic X-ray line is related to the atomic number $Z$ of the target material by:
   $\sqrt{\nu} = a(Z - b)$
   Since the wavelength $\lambda$ is inversely proportional to frequency ($\lambda = \frac{c}{\nu}$):
   $\lambda = \frac{c}{a^2(Z - b)^2}$
   As the atomic number $Z$ increases, the energy difference between the inner shells increases, resulting in an increased frequency $\nu$ and a decreased wavelength $\lambda$.
   Therefore, statement (A) is correct.

2. Analysis of the Cut-off Wavelength of Continuous X-rays (Options B and D):
   The continuous spectrum of X-rays (Bremsstrahlung) is produced when high-energy electrons are decelerated by the target material. The maximum photon energy corresponds to an electron giving up all its kinetic energy in a single collision:
   $E_{\max} = h\nu_{\max} = \frac{hc}{\lambda_{\min}} = eV$
   where:

   • $h$ is Planck's constant,
   • $c$ is the speed of light,
   • $e$ is the elementary charge,
   • $V$ is the accelerating potential of the tube, and
   • $E_k = eV$ is the kinetic energy of the striking electrons.

   Solving for the minimum (cut-off) wavelength $\lambda_{\text{cut-off}}$ (or $\lambda_{\min}$):
   $\lambda_{\text{cut-off}} = \frac{hc}{eV} = \frac{hc}{E_k}$
   Thus, the cut-off wavelength depends solely on the energy of the incident electrons (accelerating potential $V$) and is independent of the nature or atomic number of the target material.
   Therefore, statement (D) is correct, while statement (B) is wrong.

3. Analysis of Intensity (Option C):
   The intensity of the characteristic X-rays depends on the rate at which target atoms are ionized, which in turn depends on the tube current (number of electrons striking per second) and the accelerating potential (energy per electron). Both of these parameters directly determine the electrical power supplied to the X-ray tube ($P = V I$).
   Therefore, statement (C) is correct.

Since the question asks to identify the statement that is WRONG, the correct choice is B.