Question

In an X-ray tube, electrons emitted from a filament (cathode) carrying current $I$ hit a target (anode) at a distance $d$ from the cathode. The target is kept at a potential $V$ higher than the cathode resulting in the emission of continuous and characteristic X-rays. If the filament current $I$ is decreased to $\frac{1}{2}$, the potential difference $V$ is increased to $2V$, and the separation distance $d$ is reduced to $\frac{d}{2}$, then

• A. The cut-off wavelength will reduce to half, and the wavelengths of the characteristic X-rays will remain the same
• B. The cut-off wavelength, as well as the wavelengths of the characteristic X-rays, will remain the same
• C. The cut-off wavelength will reduce to half, and the intensities of all the X-rays will decrease
• D. The cut-off wavelength will become two times larger, and the intensity of all the X-rays will decrease

Answer 1

Answer: (A), (C)

The cut-off wavelength will reduce to half, and the intensities of all the X-rays will decrease

Correct Options:

Option A: The cut-off wavelength will reduce to half, and the wavelengths of the characteristic X-rays will remain the same

1. The cut-off wavelength is given by the formula $\lambda =\frac{hc}{eV}$, where $h$ is the Planck's constant, $c$ is the speed of light, $e$ is the charge of an electron and $V$ is the potential difference.
2. This implies that, $\lambda \propto \frac{1}{V}$. Since all other variables are constants, when the potential difference will increase, the wavelength will decrease proportionally.
3. Thus, when potential is $2V$, the cut-off wavelength will be $\frac{\lambda }{2}$.
4. Whereas the characteristic wavelength does not depend on current, potential difference, and separation instance, so there will be no change in characteristic wavelength.

Option C: The cut-off wavelength will reduce to half, and the intensities of all the X-rays will decrease

1. The intensity is directly proportional to the number of electrons that are emitted per second, $I\propto \frac{dN}{dt}$.
2. When the filament current is reduced, the number of electrons will reduce. Thus, the intensity will also reduce.

Incorrect Options:

Option B

The characteristic wavelength will remain the same, but the cutoff will become half. Thus, this option is incorrect.

Option D

The intensity will decrease, but the cutoff will become half. Thus, this option is incorrect.

Hence, the correct options are A and C.