Question

If ${\lambda _c}$ is cut off wavelength produced in X-ray. If operating tube voltage is increased then:
(A) $\left( {{\lambda _{K\alpha }} - {\lambda _c}} \right)$ increases.
(B) $\left( {{\lambda _{K\alpha }} - {\lambda _c}} \right)$ decreases.
(C) ${\lambda _{K\alpha }}$ increases.
(D) ${\lambda _{K\alpha }}$ decreases.

Answer 1

The value of ${\lambda _{ka}}$ depends on the material properties so it doesn’t change. Therefore the value of ${\lambda _c}$ depends upon the voltage by the relation, $\lambda = \dfrac{{hc}}{E}$. So as the voltage increases, the wavelength decreases.

Formula Used: In this solution we will be using the following formula,
$\lambda = \dfrac{{hc}}{E}$ where $\lambda$ is the wavelength of the wave and $E$ is the energy
and $h$ is the Planck’s constant and $c$ is the speed of light.

Complete Step by Step Solution:
The cut-off wavelength in an X-ray varies with the operating tube voltage. The wavelength of the $K - \alpha$ line given by ${\lambda _{K\alpha }}$ is the characteristic of the material and it remains constant in this case.
So the cut-off wavelength varies with the energy as,
${\lambda _c} = \dfrac{{hc}}{E}$
Now the values of Planck’s constant and the speed of light remains constant, so the wavelength is inversely proportional to the energy and can be written as,
${\lambda _c} \propto \dfrac{1}{E}$
Since, the of the X-ray energy and the voltage of the operating tube are directly proportional, so we can write,
${\lambda _c} \propto \dfrac{1}{V}$
In the question we are told that the operating voltage increases. So since the wavelength is inversely proportional, so as the value of the voltage increases, the wavelength decreases.
Therefore, the difference between the wavelength of the $K - \alpha$ line and the cut off wavelength will increase as the voltage increases.
Therefore as the voltage increases, $\left( {{\lambda _{K\alpha }} - {\lambda _c}} \right)$ increases

Therefore the correct answer is option A.

Note: In the heavy elements when there is a vacancy formed in the K shell (n = 1) then the electrons from the upper levels drop down to fill this gap. The X-rays which are produced by the transition of an electron from the n = 2 to the n = 1 levels are called the $K - \alpha$ lines.