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 $\dfrac{1}{2}$, the potential difference V is increased to 2v, and the separation distance d is reduced to $\dfrac{d}{2}$, 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

Hint: We have the relation between energy and potential. We also know the relation between potential and wavelength. Current and intensity are directly related. From all these, we can answer this question. If we know the graph of the intensity of current with wavelength it will become even easier to answer this question.

Formula Used:
$E=\dfrac{hc}{\lambda }$
Where,
$\lambda$- Wavelength of the characteristic x-ray

Complete answer:
X-ray is one of the electromagnetic radiations of extremely short wavelengths. Characteristic X-rays are emitted when an outer-shell electron completes the vacancy in the inner shell.

First, we have the relation between energy and potential as:
Energy, E=qv
Where q is the charge. (Here we can represent “q” as “e” since the case of electrons is discussed.)

Therefore, energy (E)= eV

According to Planck’s theory, we also have:
Energy, $E=\dfrac{hc}{\lambda }$
$eV=\dfrac{hc}{\lambda }$

From this equation we can say that Potential is inversely proportional to wavelength.
$V\propto \dfrac{1}{\lambda }$

That is when potential increases twice the first value then according to the inverse relationship between wavelength and potential, wavelength decreases to half the initial value. That is, the cut-off wavelength is reduced to half.

When the distance of separation decreases, intensity also decreases. Also when the current in the filament is reduced then the intensity will also be reduced since there will be a decrease in the number of emitted electrons.

Therefore, the answer is (a, c)

Note: For some sections there can be one or more answers. You should carefully read the questions and options and select the answer. Remember even if the cut-off wavelength changes the wavelength of characteristic x-rays are the same. The wavelength of characteristic x-ray depends on the material used as the metal target or it depends only on its atomic number